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 = x², x = y²; about y = −1
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.
x = (y − 3)², x = 4; about y = 1

Answer 1

To find the volume using the method of cylindrical shells, set up the integral as V = ∫ [2π(x - 1) * (4 - (x - 3)²)] dx.

Step-by-step explanation:

To find the volume using the method of cylindrical shells, we will integrate over concentric cylindrical shells. The radius of each shell will be the distance from the specified axis to the curve, and the height will be the difference in the x-values of the curve at each end of the shell. The integral will be set up as follows:

V = ∫ [2π(x - 1) * (4 - (x - 3)²)] dx