Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.y = 8 − x², y = x²; about x = 2

Answer 1

To find the volume using the method of cylindrical shells, set up an integral that represents the sum of the volumes of all the cylindrical shells that make up the solid. Evaluate the integral to find the volume.

Step-by-step explanation:

To find the volume using the method of cylindrical shells, we need to set up an integral that represents the sum of the volumes of all the cylindrical shells that make up the solid. The radius of each shell is the distance from the axis of rotation (x = 2) to the curve y = 8 - x^2 or y = x^2, and the height of each shell is the difference between the two curves at a given x-value.

The integral is given by:

V = 2π∫(2 - x)(8 - x^2 - x^2) dx, where the limits of integration are the x-values of the intersections between the two curves.

Simplifying and evaluating the integral will give you the volume of the solid.