# Volume via Triple Integrals.

The solid enclosed by the cylinder $x^2 + y^2 = 9$ and the planes y + z = 5 and z=1.

The biggest part for me (usually) is just being able to find my limits of integration for these problems (any suggestions about that would also be greatly appreciated). I think I found the correct limits for this problem...

$$\iiint dV$$

$$\int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{1}^{5-y} dzdydx$$

$$\int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} (4-y) dydx$$

At this point I start to get lost. Should I switch it to polar coordinates? I tried to do that from the last step above and it came out wrong. Here's my first step into the polar coordinate switch...

$$\int_0^{2\pi} \int_0^1 (4-rsin\theta)rdrd\theta$$

Does this look like I'm headed in the right direction? This chapter is completely confusing me.

Related Calculus and Beyond Homework Help News on Phys.org
SammyS
Staff Emeritus
Homework Helper
Gold Member
The solid enclosed by the cylinder $x^2 + y^2 = 9$ and the planes y + z = 5 and z=1.

The biggest part for me (usually) is just being able to find my limits of integration for these problems (any suggestions about that would also be greatly appreciated). I think I found the correct limits for this problem...

$$\iiint dV$$

$$\int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{1}^{5-y} dzdydx$$

$$\int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} (4-y) dydx$$

At this point I start to get lost. Should I switch it to polar coordinates? I tried to do that from the last step above and it came out wrong.
...
Why change coordinates?

That's a basic integral.

What is $\displaystyle \int(4-y)\,dy\,?$

I'm doing integral[0,2pi], integral[0,3], integral[1,5-rsintheta] of 1 dz r dr dtheta
Sorry for the mess. Don't know how to display the integral sign. I got 36 pi.

I'm doing integral[0,2pi], integral[0,3], integral[1,5-rsintheta] of 1 dz r dr dtheta
Sorry for the mess. Don't know how to display the integral sign. I got 36 pi.
Yeah, those limits make sense. The answer is $36\pi$, so you got it. My integrals started looking insane so I figured -rightly- that I was doing something wrong. Thanks.

Have any general advice for finding the limits? That seems to be my biggest weak-point.

Why change coordinates?

That's a basic integral.

What is $\displaystyle \int(4-y)\,dy\,?$
I know that basic integral, but the limits around it make it really intimidating because you'd have to end up using trig-subs. Right? (Thanks for the help, btw)