(9/30/08)Math 10B. Lecture Examples.Section 6.2. Constructing antiderivatives analytically†Example 1 (a) Find the antiderivativeZ3√x +4x2− 3dx.(b) Check the answer by differentiation.Answer: (a)Z3√x +4x2− 3dx == 2x3/2− 4 x−1− 3 x + C(b)ddx(2x3/2− 4x−1− 3 x) = 3√x +4x2− 3Example 2 What is the value of the integralZ1−1x2dx?Answer:Z1−1x2dx =23Example 3 EvaluateZ21(4x1/3+ 6x−2) dx.Answer:Z21(4x1/3+ 6x−2) dx = 3(24/3)Example 4 Find the area of t he region bounded by the curve y = 3x2− x3andthe x-axis.Answer: Figure A3 • [Area] =274x1 2y24y = 3x2− x3Figure A3Example 5 Suppose that the t emperature in a room is 50◦F at time t = 0 (hours) andthat the rate of change of the temperature is r = 12t2− 4t3degrees perhour at time t for 0 ≤ t ≤ 2. What is t he temperature at t = 2?Answer: The temperature at t = 2 is 66◦F†Lecture notes to accompany Section 6.2 of Calculus by Hughes-Hallett et al.1Math 10B. Lecture Examples. (9/30/08) Section 6.2, p. 2Example 6 Find the area of t he region bounded by y = x2and y = 2x.Answer: Figure A5. [Area] =43x1 2y24y = x2y = 2xFigure A5Example 7 Find the area of t he region bounded by y = sin x and y = 2 for 0 ≤ x ≤12π.Answer: [Area] = π − 1Example 8 EvaluateZ521xdx andZ−2−51xdx.Answer:Z521xdx = ln(5) − ln(2) •Z−2−51xdx = ln( 2) − ln(5)Example 8 Find a formula for the function y = g(x) such that g0(x) = exfor all x andg(2 ) = 10.Answer: g(x) = ex+ 10 − e2Example 9 A car is 30 miles north of a town at time t = 0 (hours) and its velocitytoward the north is v(t) = 60 + 5 cos t + 8 sin t miles per hour for 0 ≤ t ≤ 3.Where is it at t = 3?Answer: The car is 5 sin(3) − 8 cos(3) + 218.= 226.63 miles north of the town at t = 3.Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 6.5: 1–4Section 6.7: 1–3, 8, and 9Section 7.1: 1 and 2Section 7.7: 1 and 3‡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