# How do you find the volume of the region enclosed by the curves y=x, y=-x, and x=1 rotated about the y axis?

Oct 10, 2015

If rotating about the y-axis, the shell method is your best bet to a quick solution.

#### Explanation:

$V o l u m e = 2 \pi {\int}_{0}^{1} x \left[x - \left(- x\right)\right] \mathrm{dx} = 2 \pi {\int}_{0}^{1} \left(2 {x}^{2}\right) \mathrm{dx}$

$= \frac{4}{3} \pi$

hope that helped

Oct 10, 2015

Look at the big picture then break it apart into extrusions.

#### Explanation:

https://www.desmos.com/calculator/zem7gewjxm

I see it like a (Cylinder-2Cone) instead of a funky shape. So if we find the volume of the cylinder we get $2 \pi$ because $\pi {r}^{2} h$.

Now we have 2 Cones which are $\pi {r}^{2} \frac{h}{3}$ which turns out to be $\frac{\pi}{3}$ because the height is now one. Now we take $2 \pi$-$2 \left(\frac{\pi}{3}\right)$ or a total $V = \frac{4 \pi}{3}$