Set up the integral that uses the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

y = 7x − x²,   y = 10;  about the line x = 2

Answer 1

7x - x² = 10; x² - 7x + 10 =0; x = 2 or x = 5.

Volume v = 2π∫₂⁵(x - 2)(7x - x² - 10)dx = 13.5π

Answer 2

We have two curves both functions of x. If we set them equal to each other, we will get the points of their intersection, which define the bounds of our region; thus, these are our bounds of integration. Then, we use the washer method and simply plug in the information given.