MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region S is bounded above by the curve y=x, below by the x–axis, and on the right by the vertical line x=9. The region is revolved about the vertical line x=−1, producing a solid.
(1) Indicate appropriate limits of integration for a single definite integral that will compute the volume of the solid. Justify the limits you choose.
(2) Derive a definite integral (with respect to a single variable) whose value equals the exact volume of the solid. Show algebraic steps leading to the integrand and include the limits found in part (a). Do not evaluate the integral in this part.
(3) Evaluate the integral from part (b) to obtain the exact volume of the solid. Express your answer in terms of π and simplify completely.