HH_Sec_5_2 - (9/30/08) Math 10B. Lecture Examples. Section...

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Unformatted text preview: (9/30/08) Math 10B. Lecture Examples. Section 5.2. The definite integral† 3 Example 1 Use the formula for the area of a triangle to evaluate −3 3 (x + 1) dx. Answer: Figure A1 • (x + 1) dx = 6. −3 y 4 3 2 −3 A FIigure A1 −1 −2 1 y =x+1 B 2 3 x 1 Example 2 Calculate the right Riemann sum for 0 x2 dx corresponding to the partition of [0,1] into five equal subintervals. Draw the curve y = x2 with the rectangles whose areas give the Riemann sum. Answer: Figure A2 • [Right Riemann sum] = 0.44 y 1 y = x2 Figure A2 1 x † Lecture notes to accompany Section 5.2 of Calculus by Hughes-Hallett et al. 1 Math 10B. Lecture Examples. (9/30/08) Example 3 Use the fact that the curve y = √ Section 5.2, p. 2 16 − x2 is the upper half of the circle 0 −4 x2 + y2 = 16 of radius 4 to find the exact value of 0 16 − x2 dx. Answer: Figure A3 • 16 − x2 dx = 4π −4 y y= 2 2 √ 16 − x2 x Figure A3 Example 4 Use five rectangles of equal width to find the approximate value of 50 H(x) dx for the function y = H(x) of Figure 1. 0 y 10 5 y = H ( x) 20 FIGURE 1 −5 50 50 x Answer: One answer: Figure A4 • H (x) dx ≈ (5 + 7 + 2.5 − 5 + 4)(10) = 135 0 y 10 5 y = H ( x) 20 Figure A4 −5 50 x Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 6.2: 1–4 chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. ‡ The ...
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HH_Sec_5_2 - (9/30/08) Math 10B. Lecture Examples. Section...

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