Integrating a Solid Enclosed by a Cylinder and Two Planes

[mateomy]

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...

<br /> \iiint dV<br />

<br /> \int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{1}^{5-y} dzdydx<br />

<br /> \int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} (4-y) dydx<br />

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...

<br /> \int_0^{2\pi} \int_0^1 (4-rsin\theta)rdrd\theta<br />

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

 

[SammyS]

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...

<br /> \iiint dV<br />

<br /> \int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{1}^{5-y} dzdydx<br />

<br /> \int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} (4-y) dydx<br />

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\,?

 

[eurekameh]

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.

 

[mateomy]

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.

 

[mateomy]

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)