(9/30/08)Math 10B. Lecture Examples.Section 5.2. The definite integral†Example 1 Use the formula for the area of a triangle to evaluateZ3−3(x + 1) dx.Answer: Figure A1 •Z3−3(x + 1) dx = 6.x1 2 3y−2−1234y = x + 1−3ABFIigure A1Example 2 Calculate the right Riemann sum forZ10x2dx co rresponding to thepartition of [0,1] into five equal subintervals. Draw the curve y = x2withthe rectangles whose areas give the Riemann sum.Answer: Figure A2 • [Right Riemann sum] = 0.44x1y1y = x2Figure A2†Lecture notes to accompany Section 5.2 of C alculus by Hughes-Hallett et al.1Math 10B. Lecture Examples. (9/30/08) Section 5.2, p. 2Example 3 Use the fact that the curve y =√16 − x2is the upper half of the circlex2+ y2= 1 6 of radius 4 to find the exact value ofZ0−4p16 − x2dx.Answer: Figure A3 •Z0−4p16 − x2dx = 4πx2y2y =√16 − x2Figure A3Example 4 Use five rectangles of equal width to find t he approximate value ofZ500H(x) dx for the function y = H(x) of Figure 1.x5020y−5510y = H(x)FIGURE 1Answer: One answer: Figure A4 •Z500H(x) dx ≈ (5 + 7 + 2.5 − 5 + 4)(10) = 135x5020y−5510y = H(x)Figure A4Inte ractive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 6.2: 1–4‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the