.                 Sections 5.1–5.2               Areas and Distances               The Definite Integral                  ...
Announcements         Final December 20,         12:00–1:50pm                cumulative                location TBD       ...
Objectives from Section 5.1         Compute the area of a         region by approximating it         with rectangles and l...
Objectives from Section 5.2         Compute the definite         integral using a limit of         Riemann sums         Es...
OutlineArea through the Centuries   Euclid   Archimedes   CavalieriGeneralizing Cavalieri’s method  AnalogiesDistances   O...
Easy Areas: RectangleDefinitionThe area of a rectangle with dimensions ℓ and w is the product A = ℓw.                     ...
Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle.                            ...
Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle.                            ...
Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle.                            ...
Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle.                            ...
Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle.                            ...
Easy Areas: TriangleBy copying and pasting, a triangle can be made into a parallelogram.                                  ...
Easy Areas: TriangleBy copying and pasting, a triangle can be made into a parallelogram.                                  ...
Easy Areas: TriangleBy copying and pasting, a triangle can be made into a parallelogram.                                  ...
Easy Areas: Other PolygonsAny polygon can be triangulated, so its area can be found by summingthe areas of the triangles: ...
Hard Areas: Curved Regions                           .???                                                                 ...
Meet the mathematician: Archimedes       Greek (Syracuse), 287 BC       – 212 BC (after Euclid)       Geometer       Weapo...
Meet the mathematician: Archimedes       Greek (Syracuse), 287 BC       – 212 BC (after Euclid)       Geometer       Weapo...
Meet the mathematician: Archimedes       Greek (Syracuse), 287 BC       – 212 BC (after Euclid)       Geometer       Weapo...
Archimedes and the Parabola                                                        .Archimedes found areas of a sequence o...
Archimedes and the Parabola                                                      1                                        ...
Archimedes and the Parabola                                                      1                                  1     ...
Archimedes and the Parabola                          1                                                          1         ...
Archimedes and the Parabola                          1                                                           1        ...
Summing the seriesWe would then need to know the value of the series                        1   1           1             ...
Summing the seriesWe would then need to know the value of the series                        1   1           1             ...
Summing the seriesWe would then need to know the value of the series                        1   1           1             ...
Summing the seriesWe would then need to know the value of the series                        1   1           1             ...
Cavalieri       Italian,       1598–1647       Revisited the       area       problem with       a different       perspec...
Cavalieris method                                                    Divide up the interval into                          ...
Cavalieris method                                                    Divide up the interval into                          ...
Cavalieris method                                                      Divide up the interval into                        ...
Cavalieris method                                                      Divide up the interval into                        ...
Cavalieris method                                                    Divide up the interval into                          ...
Cavalieris method                                                    Divide up the interval into                          ...
Cavalieris method                                                        Divide up the interval into                      ...
Cavalieris method                                                        Divide up the interval into                      ...
Cavalieris method                                                    Divide up the interval into                          ...
What is Ln ?                                                                                               1Divide the int...
What is Ln ?                                                             1Divide the interval [0, 1] into n pieces. Then e...
What is Ln ?                                                             1Divide the interval [0, 1] into n pieces. Then e...
What is Ln ?                                                             1Divide the interval [0, 1] into n pieces. Then e...
What is Ln ?                                                             1Divide the interval [0, 1] into n pieces. Then e...
Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have                     ( )            ( ...
Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have                     ( )            ( ...
Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have                     ( )            ( ...
Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have                     ( )            ( ...
Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have                     ( )            ( ...
Cavalieris method with different heights                                                               1 13 1 23          ...
Cavalieris method with different heights                                                               1 13 1 23          ...
OutlineArea through the Centuries   Euclid   Archimedes   CavalieriGeneralizing Cavalieri’s method  AnalogiesDistances   O...
Cavalieris method in generalLet f be a positive function defined on the interval [a, b]. We want tofind the area between x...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.left endpoint...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. right endpoi...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.  midpoints… ...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.  the minimum...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.   the maximu...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.     …even ra...
Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.     …even ra...
Theorem of the DayTheoremIf f is a continuous function on [a, b]or has finitely many jumpdiscontinuities, then            ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]or has finitely many jumpdiscontinuities, then            ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L1 = 3.0or has ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L2 = 5.25or has...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L3 = 6.0or has ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L4 = 6.375or ha...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L5 = 6.59988or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L6 = 6.75or has...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L7 = 6.85692or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L8 = 6.9375or h...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L9 = 6.99985or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L10 = 7.04958or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L11 = 7.09064or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L12 = 7.125or h...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L13 = 7.15332or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L14 = 7.17819or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L15 = 7.19977or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L16 = 7.21875or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L17 = 7.23508or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L18 = 7.24927or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L19 = 7.26228or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L20 = 7.27443or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L21 = 7.28532or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L22 = 7.29448or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L23 = 7.30406or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L24 = 7.3125or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L25 = 7.31944or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L26 = 7.32559or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L27 = 7.33199or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L28 = 7.33798or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L29 = 7.34372or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           L30 = 7.34882or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R1 = 12.0or has...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R2 = 9.75or has...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R3 = 9.0or has ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R4 = 8.625or ha...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R5 = 8.39969or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R6 = 8.25or has...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R7 = 8.14236or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R8 = 8.0625or h...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R9 = 7.99974or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R10 = 7.94933or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R11 = 7.90868or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R12 = 7.875or h...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R13 = 7.84541or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R14 = 7.8209or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R15 = 7.7997or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R16 = 7.78125or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R17 = 7.76443or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R18 = 7.74907or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R19 = 7.73572or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R20 = 7.7243or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R21 = 7.7138or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R22 = 7.70335or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R23 = 7.69531or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R24 = 7.6875or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R25 = 7.67934or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R26 = 7.6715or ...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R27 = 7.66508or...
Theorem of the DayTheoremIf f is a continuous function on [a, b]                                           R28 = 7.6592or ...
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
Lesson 24: Areas, Distances, the Integral (Section 021 slides
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We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.

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Lesson 24: Areas, Distances, the Integral (Section 021 slides

  1. 1. . Sections 5.1–5.2 Areas and Distances The Definite Integral V63.0121.021, Calculus I New York University December 2, 2010 Announcements Final December 20, 12:00–1:50pm . . . . . .
  2. 2. Announcements Final December 20, 12:00–1:50pm cumulative location TBD old exams on common website . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 2 / 56
  3. 3. Objectives from Section 5.1 Compute the area of a region by approximating it with rectangles and letting the size of the rectangles tend to zero. Compute the total distance traveled by a particle by approximating it as distance = (rate)(time) and letting the time intervals over which one approximates tend to zero. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 3 / 56
  4. 4. Objectives from Section 5.2 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. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 4 / 56
  5. 5. OutlineArea through the Centuries Euclid Archimedes CavalieriGeneralizing Cavalieri’s method AnalogiesDistances Other applicationsThe definite integral as a limitEstimating the Definite IntegralProperties of the integralComparison Properties of the Integral . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 5 / 56
  6. 6. Easy Areas: RectangleDefinitionThe area of a rectangle with dimensions ℓ and w is the product A = ℓw. w . ℓIt may seem strange that this is a definition and not a theorem but wehave to start somewhere. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 6 / 56
  7. 7. Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle. . b . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 7 / 56
  8. 8. Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle. h . b . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 7 / 56
  9. 9. Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle. h . . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 7 / 56
  10. 10. Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle. h . b . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 7 / 56
  11. 11. Easy Areas: ParallelogramBy cutting and pasting, a parallelogram can be made into a rectangle. h . bSoFactThe area of a parallelogram of base width b and height h is A = bh . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 7 / 56
  12. 12. Easy Areas: TriangleBy copying and pasting, a triangle can be made into a parallelogram. . b . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 8 / 56
  13. 13. Easy Areas: TriangleBy copying and pasting, a triangle can be made into a parallelogram. h . b . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 8 / 56
  14. 14. Easy Areas: TriangleBy copying and pasting, a triangle can be made into a parallelogram. h . bSoFactThe area of a triangle of base width b and height h is 1 A= bh 2 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 8 / 56
  15. 15. Easy Areas: Other PolygonsAny polygon can be triangulated, so its area can be found by summingthe areas of the triangles: . . . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 9 / 56
  16. 16. Hard Areas: Curved Regions .??? . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 10 / 56
  17. 17. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (after Euclid) Geometer Weapons engineer . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 11 / 56
  18. 18. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (after Euclid) Geometer Weapons engineer . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 11 / 56
  19. 19. Meet the mathematician: Archimedes Greek (Syracuse), 287 BC – 212 BC (after Euclid) Geometer Weapons engineer . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 11 / 56
  20. 20. Archimedes and the Parabola .Archimedes found areas of a sequence of triangles inscribed in aparabola. A= . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 12 / 56
  21. 21. Archimedes and the Parabola 1 .Archimedes found areas of a sequence of triangles inscribed in aparabola. A=1 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 12 / 56
  22. 22. Archimedes and the Parabola 1 1 1 8 8 .Archimedes found areas of a sequence of triangles inscribed in aparabola. 1 A=1+2· 8 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 12 / 56
  23. 23. Archimedes and the Parabola 1 1 64 64 1 1 1 8 8 1 1 64 64 .Archimedes found areas of a sequence of triangles inscribed in aparabola. 1 1 A=1+2· +4· + ··· 8 64 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 12 / 56
  24. 24. Archimedes and the Parabola 1 1 64 64 1 1 1 8 8 1 1 64 64 .Archimedes found areas of a sequence of triangles inscribed in aparabola. 1 1 A=1+2· +4· + ··· 8 64 1 1 1 =1+ + + ··· + n + ··· 4 16 4 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 12 / 56
  25. 25. Summing the seriesWe would then need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 13 / 56
  26. 26. Summing the seriesWe would then need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4FactFor any number r and any positive integer n, (1 − r)(1 + r + · · · + rn ) = 1 − rn+1So 1 − rn+1 1 + r + · · · + rn = 1−r . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 13 / 56
  27. 27. Summing the seriesWe would then need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4FactFor any number r and any positive integer n, (1 − r)(1 + r + · · · + rn ) = 1 − rn+1So 1 − rn+1 1 + r + · · · + rn = 1−rTherefore 1 1 1 1 − (1/4)n+1 1+ + + ··· + n = 4 16 4 1 − 1/4 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 13 / 56
  28. 28. Summing the seriesWe would then need to know the value of the series 1 1 1 1+ + + ··· + n + ··· 4 16 4FactFor any number r and any positive integer n, (1 − r)(1 + r + · · · + rn ) = 1 − rn+1So 1 − rn+1 1 + r + · · · + rn = 1−rTherefore 1 1 1 1 − (1/4)n+1 1 4 1+ + + ··· + n = → 3 = as n → ∞. 4 16 4 1− 1/4 /4 3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 13 / 56
  29. 29. Cavalieri Italian, 1598–1647 Revisited the area problem with a different perspective . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 14 / 56
  30. 30. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: . 0 1 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  31. 31. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 . 0 1 1 2 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  32. 32. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 L3 = . 0 1 2 1 3 3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  33. 33. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 . 0 1 2 1 3 3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  34. 34. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 L4 = . 0 1 2 3 1 4 4 4 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  35. 35. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 0 1 2 3 1 4 4 4 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  36. 36. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 0 1 2 3 4 1 L5 = 5 5 5 5 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  37. 37. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 1 4 9 16 30 0 1 2 3 4 1 L5 = + + + = 125 125 125 125 125 5 5 5 5 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  38. 38. Cavalieris method Divide up the interval into 2 y=x pieces and measure the area of the inscribed rectangles: 1 L2 = 8 1 4 5 L3 = + = 27 27 27 1 4 9 14 L4 = + + = . 64 64 64 64 1 4 9 16 30 0 1 L5 = + + + = 125 125 125 125 125 Ln =? . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 15 / 56
  39. 39. What is Ln ? 1Divide the interval [0, 1] into n pieces. Then each has width . n . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 16 / 56
  40. 40. What is Ln ? 1Divide the interval [0, 1] into n pieces. Then each has width . The nrectangle over the ith interval and under the parabola has area ( ) 1 i − 1 2 (i − 1)2 · = . n n n3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 16 / 56
  41. 41. What is Ln ? 1Divide the interval [0, 1] into n pieces. Then each has width . The nrectangle over the ith interval and under the parabola has area ( ) 1 i − 1 2 (i − 1)2 · = . n n n3So 1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2 Ln = + 3 + ··· + = n3 n n3 n3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 16 / 56
  42. 42. What is Ln ? 1Divide the interval [0, 1] into n pieces. Then each has width . The nrectangle over the ith interval and under the parabola has area ( ) 1 i − 1 2 (i − 1)2 · = . n n n3So 1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2 Ln = + 3 + ··· + = n3 n n3 n3The Arabs knew that n(n − 1)(2n − 1) 1 + 22 + 32 + · · · + (n − 1)2 = 6So n(n − 1)(2n − 1) Ln = 6n3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 16 / 56
  43. 43. What is Ln ? 1Divide the interval [0, 1] into n pieces. Then each has width . The nrectangle over the ith interval and under the parabola has area ( ) 1 i − 1 2 (i − 1)2 · = . n n n3So 1 22 (n − 1)2 1 + 22 + 32 + · · · + (n − 1)2 Ln = + 3 + ··· + = n3 n n3 n3The Arabs knew that n(n − 1)(2n − 1) 1 + 22 + 32 + · · · + (n − 1)2 = 6So n(n − 1)(2n − 1) 1 Ln = 3 → 6n 3as n → ∞. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 16 / 56
  44. 44. Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 17 / 56
  45. 45. Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 17 / 56
  46. 46. Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 3 3 1 + 2 + 3 + · · · + (n − 1)3 = n4 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 17 / 56
  47. 47. Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 3 3 1 + 2 + 3 + · · · + (n − 1)3 = n4The formula out of the hat is [ ]2 1 + 23 + 33 + · · · + (n − 1)3 = 1 2 n(n − 1) . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 17 / 56
  48. 48. Cavalieris method for different functionsTry the same trick with f(x) = x3 . We have ( ) ( ) ( ) 1 1 1 2 1 n−1 Ln = · f + ·f + ··· + · f n n n n n n 1 1 1 23 1 (n − 1)3 = · 3 + · 3 + ··· + · n n n n n n3 3 3 1 + 2 + 3 + · · · + (n − 1)3 = n4The formula out of the hat is [ ]2 1 + 23 + 33 + · · · + (n − 1)3 = 1 2 n(n − 1) So n2 (n − 1)2 1 Ln = → 4n4 4as n → ∞. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 17 / 56
  49. 49. Cavalieris method with different heights 1 13 1 23 1 n3 Rn = · 3 + · 3 + ··· + · 3 n n n n n n 3 3 3 1 + 2 + 3 + ··· + n 3 = n4 1 [1 ]2 = 4 2 n(n + 1) n n2 (n + 1)2 1 = → 4n4 4 . as n → ∞. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 18 / 56
  50. 50. Cavalieris method with different heights 1 13 1 23 1 n3 Rn = · 3 + · 3 + ··· + · 3 n n n n n n 3 3 3 1 + 2 + 3 + ··· + n 3 = n4 1 [1 ]2 = 4 2 n(n + 1) n n2 (n + 1)2 1 = → 4n4 4 . as n → ∞.So even though the rectangles overlap, we still get the same answer. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 18 / 56
  51. 51. OutlineArea through the Centuries Euclid Archimedes CavalieriGeneralizing Cavalieri’s method AnalogiesDistances Other applicationsThe definite integral as a limitEstimating the Definite IntegralProperties of the integralComparison Properties of the Integral . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 19 / 56
  52. 52. Cavalieris method in generalLet f be a positive function defined on the interval [a, b]. We want tofind 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 na 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 x0 x1 . . . xi . . .xn−1xn n . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 20 / 56
  53. 53. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval.left endpoints… ∑ n Ln = f(xi−1 )∆x i=1 . x . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  54. 54. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. right endpoints… ∑ n Rn = f(xi )∆x i=1 . x . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  55. 55. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. midpoints… ∑ ( xi−1 + xi ) n Mn = f ∆x 2 i=1 . x . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  56. 56. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. the minimum value on theinterval… . x . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  57. 57. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. the maximum value on theinterval… . x . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  58. 58. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. …even random points! . x . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  59. 59. Forming Riemann sumsWe have many choices of representative points to approximate thearea in each subinterval. …even random points! . xIn general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form theRiemann sum ∑ n Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x i=1 . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 21 / 56
  60. 60. Theorem of the DayTheoremIf f is a continuous function on [a, b]or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  61. 61. Theorem of the DayTheoremIf f is a continuous function on [a, b]or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  62. 62. Theorem of the DayTheoremIf f is a continuous function on [a, b] L1 = 3.0or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  63. 63. Theorem of the DayTheoremIf f is a continuous function on [a, b] L2 = 5.25or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  64. 64. Theorem of the DayTheoremIf f is a continuous function on [a, b] L3 = 6.0or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  65. 65. Theorem of the DayTheoremIf f is a continuous function on [a, b] L4 = 6.375or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  66. 66. Theorem of the DayTheoremIf f is a continuous function on [a, b] L5 = 6.59988or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  67. 67. Theorem of the DayTheoremIf f is a continuous function on [a, b] L6 = 6.75or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  68. 68. Theorem of the DayTheoremIf f is a continuous function on [a, b] L7 = 6.85692or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  69. 69. Theorem of the DayTheoremIf f is a continuous function on [a, b] L8 = 6.9375or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  70. 70. Theorem of the DayTheoremIf f is a continuous function on [a, b] L9 = 6.99985or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  71. 71. Theorem of the DayTheoremIf f is a continuous function on [a, b] L10 = 7.04958or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  72. 72. Theorem of the DayTheoremIf f is a continuous function on [a, b] L11 = 7.09064or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  73. 73. Theorem of the DayTheoremIf f is a continuous function on [a, b] L12 = 7.125or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  74. 74. Theorem of the DayTheoremIf f is a continuous function on [a, b] L13 = 7.15332or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  75. 75. Theorem of the DayTheoremIf f is a continuous function on [a, b] L14 = 7.17819or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  76. 76. Theorem of the DayTheoremIf f is a continuous function on [a, b] L15 = 7.19977or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  77. 77. Theorem of the DayTheoremIf f is a continuous function on [a, b] L16 = 7.21875or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  78. 78. Theorem of the DayTheoremIf f is a continuous function on [a, b] L17 = 7.23508or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  79. 79. Theorem of the DayTheoremIf f is a continuous function on [a, b] L18 = 7.24927or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  80. 80. Theorem of the DayTheoremIf f is a continuous function on [a, b] L19 = 7.26228or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  81. 81. Theorem of the DayTheoremIf f is a continuous function on [a, b] L20 = 7.27443or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  82. 82. Theorem of the DayTheoremIf f is a continuous function on [a, b] L21 = 7.28532or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  83. 83. Theorem of the DayTheoremIf f is a continuous function on [a, b] L22 = 7.29448or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  84. 84. Theorem of the DayTheoremIf f is a continuous function on [a, b] L23 = 7.30406or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  85. 85. Theorem of the DayTheoremIf f is a continuous function on [a, b] L24 = 7.3125or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  86. 86. Theorem of the DayTheoremIf f is a continuous function on [a, b] L25 = 7.31944or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  87. 87. Theorem of the DayTheoremIf f is a continuous function on [a, b] L26 = 7.32559or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  88. 88. Theorem of the DayTheoremIf f is a continuous function on [a, b] L27 = 7.33199or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  89. 89. Theorem of the DayTheoremIf f is a continuous function on [a, b] L28 = 7.33798or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  90. 90. Theorem of the DayTheoremIf f is a continuous function on [a, b] L29 = 7.34372or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  91. 91. Theorem of the DayTheoremIf f is a continuous function on [a, b] L30 = 7.34882or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. left endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  92. 92. Theorem of the DayTheoremIf f is a continuous function on [a, b] R1 = 12.0or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  93. 93. Theorem of the DayTheoremIf f is a continuous function on [a, b] R2 = 9.75or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  94. 94. Theorem of the DayTheoremIf f is a continuous function on [a, b] R3 = 9.0or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  95. 95. Theorem of the DayTheoremIf f is a continuous function on [a, b] R4 = 8.625or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  96. 96. Theorem of the DayTheoremIf f is a continuous function on [a, b] R5 = 8.39969or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  97. 97. Theorem of the DayTheoremIf f is a continuous function on [a, b] R6 = 8.25or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  98. 98. Theorem of the DayTheoremIf f is a continuous function on [a, b] R7 = 8.14236or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  99. 99. Theorem of the DayTheoremIf f is a continuous function on [a, b] R8 = 8.0625or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  100. 100. Theorem of the DayTheoremIf f is a continuous function on [a, b] R9 = 7.99974or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  101. 101. Theorem of the DayTheoremIf f is a continuous function on [a, b] R10 = 7.94933or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  102. 102. Theorem of the DayTheoremIf f is a continuous function on [a, b] R11 = 7.90868or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  103. 103. Theorem of the DayTheoremIf f is a continuous function on [a, b] R12 = 7.875or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  104. 104. Theorem of the DayTheoremIf f is a continuous function on [a, b] R13 = 7.84541or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  105. 105. Theorem of the DayTheoremIf f is a continuous function on [a, b] R14 = 7.8209or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  106. 106. Theorem of the DayTheoremIf f is a continuous function on [a, b] R15 = 7.7997or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  107. 107. Theorem of the DayTheoremIf f is a continuous function on [a, b] R16 = 7.78125or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  108. 108. Theorem of the DayTheoremIf f is a continuous function on [a, b] R17 = 7.76443or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  109. 109. Theorem of the DayTheoremIf f is a continuous function on [a, b] R18 = 7.74907or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  110. 110. Theorem of the DayTheoremIf f is a continuous function on [a, b] R19 = 7.73572or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  111. 111. Theorem of the DayTheoremIf f is a continuous function on [a, b] R20 = 7.7243or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  112. 112. Theorem of the DayTheoremIf f is a continuous function on [a, b] R21 = 7.7138or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  113. 113. Theorem of the DayTheoremIf f is a continuous function on [a, b] R22 = 7.70335or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  114. 114. Theorem of the DayTheoremIf f is a continuous function on [a, b] R23 = 7.69531or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  115. 115. Theorem of the DayTheoremIf f is a continuous function on [a, b] R24 = 7.6875or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  116. 116. Theorem of the DayTheoremIf f is a continuous function on [a, b] R25 = 7.67934or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  117. 117. Theorem of the DayTheoremIf f is a continuous function on [a, b] R26 = 7.6715or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  118. 118. Theorem of the DayTheoremIf f is a continuous function on [a, b] R27 = 7.66508or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56
  119. 119. Theorem of the DayTheoremIf f is a continuous function on [a, b] R28 = 7.6592or has finitely many jumpdiscontinuities, then { n } ∑ lim Sn = lim f(ci )∆x n→∞ n→∞ i=1exists and is the same value no . xmatter what choice of ci we make. right endpoints . . . . . . V63.0121.021, Calculus I (NYU) Sections 5.1–5.2 Areas, Distances, Integral December 2, 2010 22 / 56

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