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Lesson 25: The Definite Integral

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Lesson 25: The Definite IntegralPresentation Transcript

• Section 5.2 The Definite Integral V63.0121.002.2010Su, Calculus I New York University June 17, 2010Announcements . . . . . .
• Announcements . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 2 / 32
• Objectives Compute the definite integral using a limit of Riemann sums Estimate the definite integral using a Riemann sum (e.g., Midpoint Rule) Reason with the definite integral using its elementary properties. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 3 / 32
• Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 4 / 32
• Cavalieris method in general Let f be a positive function defined on the interval [a, b]. We want to find the area between x = a, x = b, y = 0, and y = f(x). For each positive integer n, divide up the interval into n pieces. Then b−a ∆x = . For each i between 1 and n, let xi be the ith step between n a and b. So x0 = a b−a x1 = x0 + ∆x = a + n b−a x2 = x1 + ∆x = a + 2 · ... n b−a xi = a + i · ... n b−a . . . . . x . xn = a + n · =b . 0 . 1 . . . . i . . .xn−1. n x x x x n . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 5 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. left endpoints… ∑ n Ln = f(xi−1 )∆x i=1 . . . . . . . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. right endpoints… ∑ n Rn = f(xi )∆x i=1 . . . . . . . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. midpoints… ∑ ( xi−1 + xi ) n Mn = f ∆x 2 i=1 . . . . . . . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. the minimum value on the interval… . . . . . . . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. the maximum value on the interval… . . . . . . . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. …even random points! . . . . . . . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Forming Riemann sums We have many choices of representative points to approximate the area in each subinterval. …even random points! . . . . . . . . x In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the Riemann sum ∑ n Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no matter what choice of ci we make. . x . . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no matter what choice of ci we make. . x . . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 3.0 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 5.25 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 6.0 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 6.375 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 6.59988 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 6.75 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 6.85692 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 6.9375 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 6.99985 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 7.04958 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 7.09064 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 7.125 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 7.15332 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 7.17819 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.19977 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.21875 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.23508 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.24927 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.26228 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.27443 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.28532 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.29448 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.30406 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.3125 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.31944 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.32559 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.33199 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.33798 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.34372 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.34882 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. l . .eft endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 12.0 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 9.75 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 9.0 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 8.625 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 8.39969 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 8.25 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 8.14236 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 8.0625 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 7.99974 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 7.94933 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 7.90868 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 7.875 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 7.84541 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 7.8209 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.7997 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.78125 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.76443 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.74907 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.73572 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.7243 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.7138 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.70335 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.69531 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.6875 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.67934 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.6715 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.66508 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.6592 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.65388 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.64864 R or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. r . . ight endpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 7.4998 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 7.4996 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 7.49977 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 7.49947 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 7.49966 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 7.49937 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 7.49954 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.49968 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.49988 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.49974 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.49916 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.49898 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.4994 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.49951 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.49889 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.49962 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.5 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.49939 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.49847 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.4985 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.4986 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.49878 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.49872 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 12.0 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 10.55685 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 10.0379 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 9.41515 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 8.96004 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 8.76895 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 8.6033 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 8.45757 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 8.34564 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 8.27084 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 8.20132 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 8.13838 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 8.0916 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 8.05139 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 8.01364 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.98056 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.9539 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.92815 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.90414 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.88504 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.86737 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.84958 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.83463 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.82187 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.80824 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.79504 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.78429 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.77443 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.76495 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.7558 U or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . aximum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 1 = 3.0 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 2 = 4.44312 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 3 = 4.96208 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 4 = 5.58484 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 5 = 6.0395 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 6 = 6.23103 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 7 = 6.39577 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 8 = 6.54242 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 9 = 6.65381 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 10 = 6.72797 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 11 = 6.7979 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 12 = 6.8616 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 13 = 6.90704 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 14 = 6.94762 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 6.98575 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 16 = 7.01942 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 17 = 7.04536 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 18 = 7.07005 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 19 = 7.09364 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 20 = 7.1136 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 21 = 7.13155 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 22 = 7.14804 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 23 = 7.16441 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 24 = 7.17812 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 25 = 7.19025 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 26 = 7.2019 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 27 = 7.21265 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 28 = 7.22269 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 29 = 7.23251 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 30 = 7.24162 L or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . inimum points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
• Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 8 / 32
• The definite integral as a limit Definition If f is a function defined on [a, b], the definite integral of f from a to b is the number ∫ b ∑n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 9 / 32
• Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
• Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
• Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
• Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
• Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an infinitesimal? a variable?) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
• Notation/Terminology ∫ b ∑ n f(x) dx = lim f(ci ) ∆x a ∆x→0 i=1 ∫ — integral sign (swoopy S) f(x) — integrand a and b — limits of integration (a is the lower limit and b the upper limit) dx — ??? (a parenthesis? an infinitesimal? a variable?) The process of computing an integral is called integration or quadrature . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
• The limit can be simplified Theorem If f is continuous on [a, b] or if f has only finitely many jump discontinuities, then f is integrable on [a, b]; that is, the definite integral ∫ b f(x) dx exists. a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 11 / 32
• The limit can be simplified Theorem If f is continuous on [a, b] or if f has only finitely many jump discontinuities, then f is integrable on [a, b]; that is, the definite integral ∫ b f(x) dx exists. a Theorem If f is integrable on [a, b] then ∫ b ∑ n f(x) dx = lim f(xi )∆x, a n→∞ i=1 where b−a ∆x = and xi = a + i ∆x n . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 11 / 32
• Example: Integral of x Example ∫ 3 Find x dx 0 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 12 / 32
• Example: Integral of x Example ∫ 3 Find x dx 0 Solution 3 3i For any n we have ∆x = and xi = . So n n ∑ n ∑ ( 3i ) ( 3 ) n 9 ∑ n Rn = f(xi ) ∆x = = 2 i n n n i=1 i=1 i=1 9 n(n + 1) 9 = 2· −→ n 2 2 ∫ 3 9 So x dx = = 4.5 0 2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 12 / 32
• Example: Integral of x2 Example ∫ 3 Find x2 dx 0 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 13 / 32
• Example: Integral of x2 Example ∫ 3 Find x2 dx 0 Solution 3 3i For any n we have ∆x = and xi = . So n n ∑ n ∑ ( 3i )2 ( 3 ) 27 ∑ n n Rn = f(xi ) ∆x = = 3 i2 n n n i=1 i=1 i=1 27 n(n + 1)(2n + 1) 27 = 3 · −→ =9 n 6 3 ∫ 3 So x2 dx = 9 0 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 13 / 32
• Example: Integral of x3 Example ∫ 3 Find x3 dx 0 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 14 / 32
• Example: Integral of x3 Example ∫ 3 Find x3 dx 0 Solution 3 3i For any n we have ∆x = and xi = . So n n ∑ n ∑ ( 3i )3 ( 3 ) 81 ∑ n n Rn = f(xi ) ∆x = = 4 i3 n n n i=1 i=1 i=1 81 n2 (n + 1)2 81 = 4 · −→ n 4 4 ∫ 3 81 So x3 dx = = 20.25 0 4 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 14 / 32
• Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 15 / 32
• Estimating the Definite Integral Example ∫ 1 4 Estimate dx using M4 . 0 1 + x2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 16 / 32
• Estimating the Definite Integral Example ∫ 1 4 Estimate dx using M4 . 0 1 + x2 Solution 1 1 3 We have x0 = 0, x1 = , x2 = , x3 = , x4 = 1. 4 2 4 1 3 5 7 So c1 = , c2 = , c3 = , c4 = . 8 8 8 8 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 16 / 32
• Estimating the Definite Integral Example ∫ 1 4 Estimate dx using M4 . 0 1 + x2 Solution 1 1 3 We have x0 = 0, x1 = , x2 = , x3 = , x4 = 1. 4 2 4 1 3 5 7 So c1 = , c2 = , c3 = , c4 = . 8 8 8 8 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 ( ) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 16 / 32
• Estimating the Definite Integral Example ∫ 1 4 Estimate dx using M4 . 0 1 + x2 Solution 1 1 3 We have x0 = 0, x1 = , x2 = , x3 = , x4 = 1. 4 2 4 1 3 5 7 So c1 = , c2 = , c3 = , c4 = . 8 8 8 8 ( ) 1 4 4 4 4 M4 = + + + 4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2 ( ) 1 4 4 4 4 = + + + 4 65/64 73/64 89/64 113/64 64 64 64 64 = + + + ≈ 3.1468 65 73 89 113 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 16 / 32
• Estimating the Definite Integral (Continued) Example ∫ 1 4 Estimate dx using L4 and R4 0 1 + x2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 17 / 32
• Estimating the Definite Integral (Continued) Example ∫ 1 4 Estimate dx using L4 and R4 0 1 + x2 Answer ( ) 1 4 4 4 4 L4 = + + + 4 1 + (0)2 1 + (1/4)2 1 + (1/2)2 1 + (3/4)2 16 4 16 =1+ + + ≈ 3.38118 ( 17 5 25 ) 1 4 4 4 4 R4 = + + + 4 1 + (1/4)2 1 + (1/2)2 1 + (3/4)2 1 + (1)2 16 4 16 1 = + + + ≈ 2.88118 17 5 25 2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 17 / 32
• Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 18 / 32
• Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫ b 1. c dx = c(b − a) a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
• Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫ b 1. c dx = c(b − a) a ∫ b ∫ b ∫ b 2. [f(x) + g(x)] dx = f(x) dx + g(x) dx. a a a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
• Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫ b 1. c dx = c(b − a) a ∫ b ∫ b ∫ b 2. [f(x) + g(x)] dx = f(x) dx + g(x) dx. a a a ∫ b ∫ b 3. cf(x) dx = c f(x) dx. a a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
• Properties of the integral Theorem (Additive Properties of the Integral) Let f and g be integrable functions on [a, b] and c a constant. Then ∫ b 1. c dx = c(b − a) a ∫ b ∫ b ∫ b 2. [f(x) + g(x)] dx = f(x) dx + g(x) dx. a a a ∫ b ∫ b 3. cf(x) dx = c f(x) dx. a a ∫ b ∫ b ∫ b 4. [f(x) − g(x)] dx = f(x) dx − g(x) dx. a a a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
• Proofs Proofs. When integrating a constant function c, each Riemann sum equals c(b − a). A Riemann sum for f + g equals a Riemann sum for f plus a Riemann sum for g. Using the sum rule for limits, the integral of a sum is the sum of the integrals. Ditto for constant multiples Ditto for differences . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 20 / 32
• Example ∫ 3( ) Find x3 − 4.5x2 + 5.5x + 1 dx 0 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 21 / 32
• Example ∫ 3( ) Find x3 − 4.5x2 + 5.5x + 1 dx 0 Solution ∫ 3 (x3 −4.5x2 + 5.5x + 1) dx 0 ∫ 3 ∫ 3 ∫ 3 ∫ 3 = x dx − 4.5 3 2 x dx + 5.5 x dx + 1 dx 0 0 0 0 = 20.25 − 4.5 · 9 + 5.5 · 4.5 + 3 · 1 = 7.5 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 21 / 32
• Example ∫ 3( ) Find x3 − 4.5x2 + 5.5x + 1 dx 0 Solution ∫ 3 (x3 −4.5x2 + 5.5x + 1) dx 0 ∫ 3 ∫ 3 ∫ 3 ∫ 3 = x dx − 4.5 3 2 x dx + 5.5 x dx + 1 dx 0 0 0 0 = 20.25 − 4.5 · 9 + 5.5 · 4.5 + 3 · 1 = 7.5 (This is the function we were estimating the integral of before) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 21 / 32
• Theorem of the (previous) Day Theorem If f is a continuous function on [a, b] . . 15 = 7.49968 M or has finitely many jump discontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1 exists and is the same value no . x . matter what choice of ci we make. m . . idpoints . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 22 / 32
• More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 23 / 32
• More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a ∫ a f(x) dx = 0 a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 23 / 32
• More Properties of the Integral Conventions: ∫ ∫ a b f(x) dx = − f(x) dx b a ∫ a f(x) dx = 0 a This allows us to have ∫ c ∫ b ∫ c 5. f(x) dx = f(x) dx + f(x) dx for all a, b, and c. a a b . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 23 / 32
• Example Suppose f and g are functions with ∫ 4 f(x) dx = 4 0 ∫ 5 f(x) dx = 7 0 ∫ 5 g(x) dx = 3. 0 Find ∫ 5 (a) [2f(x) − g(x)] dx 0 ∫ 5 (b) f(x) dx. 4 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 24 / 32
• Solution We have (a) ∫ 5 ∫ 5 ∫ 5 [2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx 0 0 0 = 2 · 7 − 3 = 11 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 25 / 32
• Solution We have (a) ∫ 5 ∫ 5 ∫ 5 [2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx 0 0 0 = 2 · 7 − 3 = 11 (b) ∫ 5 ∫ 5 ∫ 4 f(x) dx = f(x) dx − f(x) dx 4 0 0 =7−4=3 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 25 / 32
• Outline Recall The definite integral as a limit Estimating the Definite Integral Properties of the integral Comparison Properties of the Integral . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 26 / 32
• Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
• Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
• Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a 7. If f(x) ≥ g(x) for all x in [a, b], then ∫ b ∫ b f(x) dx ≥ g(x) dx a a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
• Comparison Properties of the Integral Theorem Let f and g be integrable functions on [a, b]. 6. If f(x) ≥ 0 for all x in [a, b], then ∫ b f(x) dx ≥ 0 a 7. If f(x) ≥ g(x) for all x in [a, b], then ∫ b ∫ b f(x) dx ≥ g(x) dx a a 8. If m ≤ f(x) ≤ M for all x in [a, b], then ∫ b m(b − a) ≤ f(x) dx ≤ M(b − a) a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
• The integral of a nonnegative function is nonnegative Proof. If f(x) ≥ 0 for all x in [a, b], then for any number of divisions n and choice of sample points {ci }: ∑ n ∑ n Sn = f(ci ) ∆x ≥ 0 · ∆x = 0 i=1 i=1 ≥0 Since Sn ≥ 0 for all n, the limit of {Sn } is nonnegative, too: ∫ b f(x) dx = lim Sn ≥ 0 a n→∞ ≥0 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 28 / 32
• The definite integral is “increasing" Proof. Let h(x) = f(x) − g(x). If f(x) ≥ g(x) for all x in [a, b], then h(x) ≥ 0 for all x in [a, b]. So by the previous property ∫ b h(x) dx ≥ 0 a This means that ∫ b ∫ b ∫ b ∫ b f(x) dx − g(x) dx = (f(x) − g(x)) dx = h(x) dx ≥ 0 a a a a So ∫ ∫ b b f(x) dx ≥ g(x) dx a a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 29 / 32
• Bounding the integral using bounds of the function Proof. If m ≤ f(x) ≤ M on for all x in [a, b], then by the previous property ∫ b ∫ b ∫ b m dx ≤ f(x) dx ≤ M dx a a a By Property ??, the integral of a constant function is the product of the constant and the width of the interval. So: ∫ b m(b − a) ≤ f(x) dx ≤ M(b − a) a . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 30 / 32
• Example ∫ 2 1 Estimate dx using the comparison properties. 1 x . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 31 / 32
• Example ∫ 2 1 Estimate dx using the comparison properties. 1 x Solution Since 1 1 ≤x≤ 2 1 for all x in [1, 2], we have ∫ 2 1 1 ·1≤ dx ≤ 1 · 1 2 1 x . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 31 / 32
• Summary The definite integral is a limit of Riemann Sums The definite integral can be estimated with Riemann Sums The definite integral can be distributed across sums and constant multiples of functions The definite integral can be bounded using bounds for the function . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 32 / 32