Illustrate your discussion with an example For single-variable definite Integrals, the tech-nique of substitution was generally used to simplify theintegrand. Discuss some motivations for using a changeof variables In a multiple Integral.50.WritingSuppose that the boundary curves of a regionRI n thexy-plane can be described as level curves of vari-ous functions. Discuss how this Information can be used to choose an appropriate change of variables for a doubleintegra overR. Illustrate your discussion with an example Suppose thatfis a nonnegative function defined on[0,1]such that the area between the graph offand the interval[0, 1]isAsand such that the area of the regionRbetweenthe graph ofg(x)=f(x:)and the interval[0,1]isAz. Ineach part, express your answer in terms ofAiandAz.(a) What is the volume of the solid of revolution generatedby revolvingRabout they-axis?(b) Find a value ofasuch that if thexy-plane were horizon-tal, the regionRwould balance on the linex=a.2.A water tank has the shape of a conical frustum with radiusof the base 5 ft, radius of the top 10 it and (vertical) height15 ft. Suppose the tank is filled with water and consider theproblem of finding the work required to pump all the waterout through a hole in the top of the tank.(e) Solve this problem using the method of Example 5 inSection 6.6.(b) Solve this problem using Definition 6.6.3. [Hint: Thinkof the base as the head of a piston that expands to a water-tight fit against the sides of the tank as the piston is pushedupward. What important result about water pressure doyou need to use?1 3.A disk of radiusais an inhomogeneous lamina whose den-sity is a functionf(r) of the distancerto the center of thelamina. Modify the argument used to derive the method ofcylindrical shells to find a formula for the mass of the lamina.4.Compare Formula (10) in Section 6.7 with Formula Illustrate your discussion with an example Let R be the region enclosed by the curvesy=x2+4,y=x, and they-axis. Find and evaluate a definite inte-gral that represents the volume of the solid generated byrevolvingRabout thex-axis.10.A football has the shape of the solid generated by revolv-ing the region bounded between thex-axis and the parabolay=4R(x:-14:)/Leabout thex-axis. Find its volume.11. Find the volume of the solid whose base is the region bounded between the curvesy=\\xandy=1/Wxfor1sxs4 and whose cross sections perpendicular to thex-axis are squares