1. ## Cylindrical coordinates..

Use cylindrical coordinates to find the volume of the solid in the first octant inside of the sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 2x

Note: set up the integral only, do not evaluate.

2. Originally Posted by boousaf
Use cylindrical coordinates to find the volume of the solid in the first octant inside of the sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 2x

Note: set up the integral only, do not evaluate.
The region they have in common is the cross sectional area of the cylinder $(x-1)^2+y^2=1$, thus, we see that $0\leq x\leq 2$, $-\sqrt{1-(x-1)^2}\leq y\leq \sqrt{1-(x-1)^2}$, and the height of the solid would be defined by $-\sqrt{4-x^2-y^2}\leq z\leq \sqrt{4-x^2-y^2}$

In cylindrical coordinates, we can see that $1\leq r\leq 2$, $0\leq\vartheta\leq2\pi$, and $-\sqrt{4-r^2}\leq z\leq \sqrt{4-r^2}$

After a little more work, we see that the volume could be represented as $2\int_0^{2\pi}\int_1^2 \int_0^{\sqrt{4-r^2}}r\,dz\,dr\,d\vartheta$

Does this make sense?

[I'm pretty sure my limits on $r$ are correct...]

--Chris

3. Thanks very much Chris! Perfect explanation!