1. ## Cylindrical shells revolved about the line y=1

Use cylindrical shells to find the volume of the solid that is generate when the region that enclosed by $\displaystyle y=x^3$, $\displaystyle y=1$, $\displaystyle x=0$ is revolved about the line $\displaystyle y=1$

As it said "cylindrical shells" to find the volume.

I transform $\displaystyle y=x^3$ to $\displaystyle x=\sqrt[3]{y}$

The area of this surface is $\displaystyle 2\pi x f(x)$, and then integrate from a to b to get cylindrical shells formula. For this problem, I changed a to b to c to d. And $\displaystyle 2\pi x f(x)$ to $\displaystyle 2\pi y f(y)$

My problem is to be revolved about the line y=1, that means it similar to revolve about x-axis, but higher 1 unit. I don't know how to calculate it.

I don't know how to find y and f(y).

2. Am I correct, if the answer is $\displaystyle 2\pi$ by using $\displaystyle 2\pi (y+1)(1-\sqrt[3]{y})$ as area.
$\displaystyle 2{\pi}\int_{0}^{1}(1-x)x^{3}dx$