MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
Let R be the region bounded by y=2x−x2 and the x-axis. The region R is revolved about the vertical line x=−1.
(a) Determine the limits of integration and write a single integral expression for the volume of the resulting solid that does not require solving for x as a function of y. Explain how your choice of variable of integration avoids that inversion.
(b) Evaluate the integral from part (a) to find the volume of the solid, expressing your answer in terms of π.
(c) Suppose instead that the region R is revolved about the horizontal line y=−3. Without performing any calculations, state whether using x or y as the variable of integration would now be more efficient, and justify your answer based on the geometry of the new solid.