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Section 15.6 : Triple Integrals in Cylindrical Coordinates

2. Evaluate Eex2z2dV where E is the region between the two cylinders x2+z2=4 and x2+z2=9 with 1y5 and z0.

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Start Solution

Okay, let’s start off with a quick sketch of the region E so we can get a feel for what we’re dealing with.

We know that x2+z2=4 and x2+z2=9 are cylinders of radius 2 and 3 respectively that are centered on the y-axis. The range 1y5 tells us that we will only have the cylinders in this range of y’s. Finally, the z0 tells us that we will only have the lower half of each of the cylinders.

Here then is the sketch of E.

We’ve given the sketches with a set of “traditional” axes as well as a set of “box” axes to help visualize the surface and region.

The “front” of E is just the portion of the plane y=5 that “caps” the front and is the orange ring in the sketch. The “back” of E is the portion of the plane y=1 that caps the back of the region and is not shown in the sketch due to the orientation of axis system.

Show Step 2

So, from the sketch above it looks like the region D will be back in the xz-plane and so we’ll need to integrate with respect to y first. In this case this is even easier because both the front and back portions of the surfaces are just the planes y=5 and y=1 respectively. That means that the y limits are,

1y5 Show Step 3

Now, let’s think about the D for this problem. If we look at the object from along the y axis we see the lower half of the ring with radii 2 and 3 as shown below.

Note that the x-axis orientation is switched from the standard orientation to more accurately match what you’d see if you did look at E from along the y-axis. In other words, the positive x values are on the left side and the negative x values are on the right side. The orientation of the x-axis doesn’t change that we still see a portion of a ring that lies in the xz-plane and this is in fact the D for this problem.

Because D is in the xz-plane and it is a portion of a ring that means that we’ll need to use the following “modified” version of cylindrical coordinates.

x=rcosθy=yz=rsinθ

This also matches up with the fact that we need to integrate y first (as we determined in Step 2) and the first variable of integration with cylindrical coordinates is always the “free” variable (i.e. not the one involving the trig functions).

So, we can easily describe the ring in terms of r and θ so here are the cylindrical coordinates for this problem.

πθ2π2r31y5 Show Step 4

Plugging these limits into the integral and converting to cylindrical coordinates gives,

Eex2z2dV=2ππ3251rer2dydrdθ

Don’t forget x2+z2=r2 under our modified cylindrical coordinates and also don’t forget that dV=rdydrdθ and so we pick up another r when converting the dV to cylindrical coordinates (that will be very helpful with the r integration).

Show Step 5

Okay, now all we need to do is evaluate the integral. Here is the y integration.

Eex2z2dV=2ππ32rer2y|51drdθ=2ππ324rer2drdθ Show Step 6

Next let’s do the r integration.

Eex2z2dV=2ππ(2er2)|32dθ=2ππ2(e4e9)dθ Show Step 7

Finally, we’ll do the θ integration.

Eex2z2dV=2(e4e9)θ|2ππ=2π(e4e9)=0.1143

The trickiest part of this one was probably the sketch of E. Once you see that and how to get the D for the integral the rest of the problem was pretty simple for the most part.