1. Evaluate the integral. Show all work. More than one method may work. (a) 1 rv9- 12 (b) 19 7:2 1 (c) Vx2 + 49 (d) 1 Vx2 + 1 (e) V36-12dx (1) V1 -36x2dar. 1 (g) dx V36 + 2 1 (h) 49+ :22. A pharmaceutical company is designing a new drug whose shape in tablet form is obtained by rotating the graph of f(x) = 2(42 - 2?)1/1 on the interval [-6, 6] around the x-axis, where units are measured in millimeters. Shape of a tablet 6 - 4. -6 -4 -2 2 4 -6- As we will see, the volume of such a solid is given by " ( (()da, where a and b are the places where the graph of f(x) meets the r-axis. (a) Write down a specific definite integral that represents the amount of material, in cubic millimeters, required to fill the tablet. (b) Use trigonometric substitution to solve the definite integral and de- termine the amount of material needed.3. The main cable on a certain suspension bridge follows a parabolic curve with equation f(x) = (0.025x)2, measured in feet, as shown in the follow- ing figure: Main cable of a suspension bridge 150+ 100 50 + 500 2 XXIX As we will see, the length of a curve f(x) from = = a to r = b can be calculated from the formula Vi+ (f'(x))dx. (a) Write down a specific definite integral that represents the length of the main cable of the suspension bridge. (b) Use trigonometric substitution to solve the definite integral and de- termine the length of the cable. 4. Prove that the area of a circle of radius r is my, as follows: (a) Write down a definite integral that represents the area of the circle of radius r centered at the origin. (Hint: The equation of such a circle is a+12 = 12.) (b) Use trigonometric substitution to solve the definite integral you found in part (a). (Hint: Change the limits of integration to match your substitution.)