1. Water flows out of a large tank at a rate (in m'/hr) given by V'(t) = . If the tank initially holds 750 m? of water, when will the tank be empty? Spts . Find the areas of the regions R; and Rz (separately) shown in the attached figure: they are formed by the graphs of y = 16 x> and y = 5x 8. Spis A | + i b m 3. Evaluate the integral fuzlsinx cos 2x|dx and interpret it as the area of a region. Sketch the region. 10pts m 3. Evaluate the integral fglsinx cos 2x|dx and interpret it as the area of a region. Sketch the region. 10pts 4. The quarter circle of radius R in the first quadrant (x? + y? = R? for x = 0 and y = 0) is revolved about the x-axis to produce a hemisphere. Find the volume of the hemisphere in the following three ways: a. Apply the disk method and integrate with respect to x. 3pts b. Apply the shell method and integrate with respect to y. 3pts . Apply the general slicing method and integrate with respect to y. apis 5. The region R is bounded by the curves x = y* + 2,y = x 4, and y = 0 (see figure). a. Write a single integral that gives the area of R. Evaluate the integral. 2.5ps b. Write a single integral that gives the volume of the solid generated when R is 5. The region R is bounded by the curves x = y? + 2,y = x 4, and y = 0 (see figure). a. Write a single integral that gives the area of R. Evaluate the integral. = 25y b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis. Evaluate the integral. 2.5pts c. Write a single integral that gives the volume of the solid generated when R is revolved about the y-axis. Evaluate the integral. 2.5pts d. Suppose that S is a solid whose base is R and whose cross sections perpendicular to R and parallel to the x-axis are semicircles. Write a single integral that gives the volume of S. Evaluate it. 2.5pts 6. A spherical tank is filled with water (see figure). Find the work required to pump the water out of the spout. The density of water is 1000 kg/m?. Spts I'm 7. The velocity v of blood that flows in a blood vessel with radius R and length / at a distance r from the central axis is v(r) = 4%1 (R* r?) where P is the pressure difference between the end of the vessel and n is the viscosity of the blood. Find the average velocity over the interval 0