1. ## Solid of revolution

Find the volume of solid of revolution generated by rotating the lamina contained between $\displaystyle y=4-x^2$ and $\displaystyle y=0$, around the axis $\displaystyle x=3$.

I understand the premise of this task but can't quite arrive at the correct expression for integration, which should lead me eventually to the answer of $\displaystyle 64\pi$ (if I should believe the textbook). Any ideas?

EDIT: Sorry, corrected the typo.

2. ## Re: Solid of revolution

clarification, please ... is the region to be rotated about $\displaystyle x = 3$ in quads I and IV between $\displaystyle y = 4-x^2$ , $\displaystyle y = 4$ , and $\displaystyle x = 3$ ... ???

3. ## Re: Solid of revolution

Originally Posted by atreyyu
Find the volume of solid of revolution generated by rotating the lamina contained between $\displaystyle y=4-x^2$ and $\displaystyle y=0$, around the axis $\displaystyle x=3$.
I understand the premise of this task but can't quite arrive at the correct expression for integration, which should lead me eventually to the answer of $\displaystyle 64\pi$ (if I should believe the textbook). Any ideas?
Make this substitution $\displaystyle x=x_1+3,~y=y_1$, then

$\displaystyle y_1=4-(x_1+3)^2$ and $\displaystyle y_1=0$, around the axis $\displaystyle x_1=0$.

Find the roots $\displaystyle y_1=4-(x_1+3)^2,~y_1=0$, you must get $\displaystyle x_1=-5$ and $\displaystyle x_1=-1$

Now you can use the standard formula

$\displaystyle V=-2\pi\int\limits_{-5}^{-1}x_1(4-(x_1+3)^2)\,dx_1=\ldots=64\pi~\mathsf{(cubic~units )}$