Volume of a region bounded by a surface and planes

[forestmine]

Homework Statement

Find the volume of the region bounded by the cylinder x^2 + y^2 =4 and the planes z=0, and x+z=3.

Homework Equations

V = ∫∫∫dzdxdy

V=∫∫∫rdrdθ

The Attempt at a Solution

Alright, so I feel as though I'm missing a step somewhere along the way, but here's what I've gotten so far.

So I know that I've got a cylinder centered around the z-axis, bounded by z=0 (which would be the xy-plane) and x+z=3.

If I attempt a triple integral in Cartesian coordinates, beginning with the z limits of integration, I enter the region at z=0, and exit the region at z=3-x? I'm having a hard time picturing the x=3-x part since y=0, and so we aren't actually looking at the whole cylinder?

Well from there, I looked at the circle cast by the cylinder on the xy-plane, x^2+y^2 = 4. I solved for y for my y limits of integration, -(4-x^2)^1/2 to + (4-x^2)^1/2.

And then my x limits are simply -2 to 2.

I feel like I'm not taking into account the z=3-x for my y and x limits, though I'm not sure how I would...

Here's my attempt in cylindrical coordinates.

Since x^2+y^2 = r^2=4, I said my r limits are from r=0 to r=2.

My z limits are from z=0 to z=3-rcosθ.

And my limits for θ are from 0 to 2pi.

I feel like I'm missing something...Any help in the right direction would be greatly appreciated!

Thanks!

 

[LCKurtz]

Homework Statement

Find the volume of the region bounded by the cylinder x^2 + y^2 =4 and the planes z=0, and x+z=3.

Homework Equations

V = ∫∫∫dzdxdy

V=∫∫∫rdrdθ

The Attempt at a Solution

Alright, so I feel as though I'm missing a step somewhere along the way, but here's what I've gotten so far.

So I know that I've got a cylinder centered around the z-axis, bounded by z=0 (which would be the xy-plane) and x+z=3.

If I attempt a triple integral in Cartesian coordinates, beginning with the z limits of integration, I enter the region at z=0, and exit the region at z=3-x? I'm having a hard time picturing the x=3-x (you mean z = 3-x) part since y=0, and so we aren't actually looking at the whole cylinder?

z = 3-x is the top of the surface, so that part is OK.

Well from there, I looked at the circle cast by the cylinder on the xy-plane, x^2+y^2 = 4. I solved for y for my y limits of integration, -(4-x^2)^1/2 to + (4-x^2)^1/2.

And then my x limits are simply -2 to 2.

Yes, although after you do the dz integral you might want to use polar coordinates for the dydx integral for ease of calculation.

I feel like I'm not taking into account the z=3-x for my y and x limits, though I'm not sure how I would...

It's OK. z as a function of x was taken care of in the inside dz limits

Here's my attempt in cylindrical coordinates.

Since x^2+y^2 = r^2=4, I said my r limits are from r=0 to r=2.

My z limits are from z=0 to z=3-rcosθ.

And my limits for θ are from 0 to 2pi.

Thanks!

Your cylindrical limits are Ok as long as you are sure to integrate dz before dr because the z limits depend on r.

 

[forestmine]

Oh wow, thank you! I couldn't help but feel like I didn't quite have it.

Thanks for checking it out!