Can anyone help me with this?

Question

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by  , , and  about the -axis.

Answer 1

With the shell method, we take cross sections that are parallel to the axis of rotation.

So, take a thin vertical cross section x units from the y-axis of height x² and thickness Δx.

Rotate the cross section about the y-axis to get a shell with height x², radius x, and thickness Δx.

Volume of shell = 2πx(x²)Δx

Volume of solid = 2π∫_{(0 to 3)}x³dx = (π/2)x⁴_{(from 0 to 3)} = 81π/2