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Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
Lesson18   Double Integrals Over Rectangles Slides
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Lesson18 Double Integrals Over Rectangles Slides

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We develop double integrals for measuring volume and iterated integrals for calculating double integrals.

We develop double integrals for measuring volume and iterated integrals for calculating double integrals.

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  • 1. Section 12.1–12.2 Double Integrals over Rectangles Iterated Integrals Math 21a March 17, 2008 Announcements ◮ Office hours Tuesday, Wednesday 2–4pm SC 323 ◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b . . Image: Flickr user Cobalt123 . . . . . .
  • 2. Announcements ◮ Office hours Tuesday, Wednesday 2–4pm SC 323 ◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b . . . . . .
  • 3. Outline Last Time Double Integrals over Rectangles Recall the definite integral Definite integrals in two dimensions Iterated Integrals Partial Integration Fubini’s Theorem Average value . . . . . .
  • 4. Outline Last Time Double Integrals over Rectangles Recall the definite integral Definite integrals in two dimensions Iterated Integrals Partial Integration Fubini’s Theorem Average value . . . . . .
  • 5. Cavalieri’s method 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 nth step n between a and b. So x0 = a b−a x 1 = x 0 + ∆x = a + n b−a x 2 = x 1 + ∆x = a + 2 · n ······ b−a xi = a + i · n x x x .0 .1 .2 . i . n −1 . n xx x ······ . . . . . . . . . a b−a b . xn = a + n · =b . . n . . . .
  • 6. Forming Riemann sums We have many choices of how to approximate the area: Ln = f(x0 )∆x + f(x1 )∆x + · · · + f(xn−1 )∆x Rn = f(x1 )∆x + f(x2 )∆x + · · · + f(xn )∆x ( ) ( ) ( ) x0 + x 1 x1 + x2 x n −1 + x n Mn = f ∆x + f ∆x + · · · + f ∆x 2 2 2 . . . . . .
  • 7. Forming Riemann sums We have many choices of how to approximate the area: Ln = f(x0 )∆x + f(x1 )∆x + · · · + f(xn−1 )∆x Rn = f(x1 )∆x + f(x2 )∆x + · · · + f(xn )∆x ( ) ( ) ( ) x0 + x 1 x1 + x2 x n −1 + x n Mn = f ∆x + f ∆x + · · · + f ∆x 2 2 2 In general, choose x∗ to be a point in the ith interval [xi−1 , xi ]. Form i the Riemann sum Sn = f(x∗ )∆x + f(x∗ )∆x + · · · + f(x∗ )∆x 1 2 n ∑ n = f(x∗ )∆x i i=1 . . . . . .
  • 8. Definition The definite integral of f from a to b is the limit ∫ b ∑ n f(x) dx = lim f(x∗ )∆x i a n→∞ i=1 (The big deal is that for continuous functions this limit is the same no matter how you choose the x∗ ).i . . . . . .
  • 9. The problem Let R = [a, b] × [c, d] be a rectangle in the plane, f a positive function defined on R, and S = { (x, y, z) | a ≤ x ≤ b, c ≤ y ≤ d, 0 ≤ z ≤ f(x, y) } Our goal is to find the volume of S . . . . . .
  • 10. The strategy: Divide and conquer For each m and n, divide the interval [a, b] into m subintervals of equal width, and the interval [c, d] into n subintervals. For each i and j, form the subrectangles Rij = [xi−1 , xi ] × [yj−1 , yj ] Choose a sample point (x∗ , y∗ ) in each subrectangle and form the ij ij Riemann sum ∑∑m n Smn = f(x∗ , y∗ ) ∆A ij ij i=1 j=1 where ∆A = ∆x ∆y. . . . . . .
  • 11. Definition The double integral of f over the rectangle R is ∫∫ ∑∑ m n f(x, y) dA = lim f(x∗ , y∗ ) ∆A ij ij m,n→∞ R i=1 j=1 (Again, for continuous f this limit is the same regardless of method for choosing the sample points.) . . . . . .
  • 12. Worksheet #1 Problem Estimate the volume of the solid that lies below the surface z = xy and above the rectangle [0, 6] × [0, 4] in the xy-plane using a Riemann sum with m = 3 and n = 2. Take the sample point to be the upper right corner of each rectangle. . . . . . .
  • 13. Worksheet #1 Problem Estimate the volume of the solid that lies below the surface z = xy and above the rectangle [0, 6] × [0, 4] in the xy-plane using a Riemann sum with m = 3 and n = 2. Take the sample point to be the upper right corner of each rectangle. Answer 288 . . . . . .
  • 14. Theorem (Midpoint Rule) ∫∫ ∑∑ m n f(x, y) dA ≈ f(¯i , ¯j ) ∆A x y R i=1 j=1 where ¯i is the midpoint of [xi−1 , xi ] and ¯j is the midpoint of [yj−1 , yj ]. x y . . . . . .
  • 15. Worksheet #2 Problem Use the Midpoint Rule to evaluate the volume of the solid in Problem 1. . . . . . .
  • 16. Worksheet #2 Problem Use the Midpoint Rule to evaluate the volume of the solid in Problem 1. Answer 144 . . . . . .
  • 17. Outline Last Time Double Integrals over Rectangles Recall the definite integral Definite integrals in two dimensions Iterated Integrals Partial Integration Fubini’s Theorem Average value . . . . . .
  • 18. Partial Integration Let f be a function on a rectangle R = [a, b] × [c, d]. Then for each fixed x we have a number ∫ d A(x) = f(x, y) dy c The is a function of x, and can be integrated itself. So we have an iterated integral ∫ b ∫ b [∫ d ] A(x) dx = f(x, y) dy dx a a c . . . . . .
  • 19. Worksheet #3 Problem Calculate ∫ 3∫ 1 ∫ 1∫ 3 (1 + 4xy) dx dy and (1 + 4xy) dy dx. 1 0 0 1 . . . . . .
  • 20. Fubini’s Theorem Double integrals look hard. Iterated integrals look easy/easier. The good news is: Theorem (Fubini’s Theorem) If f is continuous on R = [a, b] × [c, d], then ∫∫ ∫ b∫ d ∫ d∫ b f(x, y) dA = f(x, y) dy dx = f(x, y) dx dy a c c a R This is also true if f is bounded on R, f is discontinuous only on a finite number of smooth curves, and the iterated integrals exist. . . . . . .
  • 21. Worksheet #4 Problem Evaluate the volume of the solid in Problem 1 by computing an iterated integral. . . . . . .
  • 22. Worksheet #4 Problem Evaluate the volume of the solid in Problem 1 by computing an iterated integral. Answer 144 . . . . . .
  • 23. Meet the mathematician: Guido Fubini ◮ Italian, 1879–1943 ◮ graduated Pisa 1900 ◮ professor in Turin, 1908–1938 ◮ escaped to US and died five years later . . . . . .
  • 24. Worksheet #5 Problem Calculate ∫∫ xy2 dA x2 + 1 R where R = [0, 1] × [−3, 3]. . . . . . .
  • 25. Worksheet #5 Problem Calculate ∫∫ xy2 dA x2 + 1 R where R = [0, 1] × [−3, 3]. Answer ln 512 = 9 ln 2 . . . . . .
  • 26. Average value ◮ One variable: If f is a function defined on [a, b], then ∫ b 1 fave = f(x) dx b−a a ◮ Two variables: If f is a function defined on a rectangle R, then ∫∫ 1 fave = f(x, y) dA Area(R) R . . . . . .
  • 27. Worksheet #6 Problem Find the average value of f(x, y) = x2 y over the rectangle R = [−1, 1] × [0, 5]. . . . . . .
  • 28. Worksheet #6 Problem Find the average value of f(x, y) = x2 y over the rectangle R = [−1, 1] × [0, 5]. Answer ∫ 5∫ 1 1 5 x2 y dx dy = 10 0 −1 6 . . . . . .

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