let r be the region bounded by y = x², x = 1, and y = 0. use the shell method to find the volume of the…

let r be the region bounded by y = x², x = 1, and y = 0. use the shell method to find the volume of the solid generated when r is revolved about the line y = - 3. set up the integral that gives the volume of the solid using the shell method. use increasing limits of integration. select the correct choice below and fill in the answer boxes to complete your choice. (type exact answers.) a. ∫ ( ) dx b. ∫ ( ) dy

let r be the region bounded by y = x², x = 1, and y = 0. use the shell method to find the volume of the solid generated when r is revolved about the line y = - 3. set up the integral that gives the volume of the solid using the shell method. use increasing limits of integration. select the correct choice below and fill in the answer boxes to complete your choice. (type exact answers.) a. ∫ ( ) dx b. ∫ ( ) dy

Answer

Explanation:

Step1: Recall shell - method formula for horizontal axis of revolution

The formula for the volume (V) using the shell method when revolving about a horizontal line (y = k) is (V=2\pi\int_{a}^{b}(r\cdot h)dx) (if integrating with respect to (x)), where (r) is the distance from the axis of revolution to a representative rectangle and (h) is the height of the representative rectangle. The region is bounded by (y = x^{2}), (x = 1), and (y = 0). The axis of revolution is (y=-3). For a vertical slice of width (dx) at a position (x) (where (0\leq x\leq1)), the radius (r=y + 3=x^{2}+3) (distance from (y=-3) to (y = x^{2})), and the height (h = 1 - x) (the horizontal extent of the slice in the (x) - direction).

Step2: Set up the integral

Using the shell - method formula (V = 2\pi\int_{a}^{b}r\cdot hdx), with (a = 0), (b = 1), (r=x^{2}+3), and (h = 1 - x), we get (V=2\pi\int_{0}^{1}(x^{2}+3)(1 - x)dx).

Answer:

A. (\int_{0}^{1}2\pi(x^{2}+3)(1 - x)dx)