Question

Consider the given curves to do the following.
y=x^3
y=0
x=1
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 1.

Answer 1

A(y)= 2*pi*y*(1-y^1/3)
\[V = 2\pi \int\limits_{0}^{1}y(1-\sqrt[3]{y}) dy\]
\[=2\pi \int\limits_{0}^{1}y - y ^{4/3} dy\]
\[=2\pi [\frac{1}{2}y ^{2} - \frac{3}{7}y ^{7/3}]\]
\[=\frac{\pi}{7}\]

Answer 2

ahh posted in wrong question lol