00:01
All right, so this question is asking us to evaluate a triple integral where e is a 3d region bounded by four surfaces, three of which are planes, one of which is a parabolic cylinder.
00:16
So now in order to do this, we have to choose our order of integration, and there are of course, there are six ways in which you can do it, but the one that i'm leaning towards is to integrate with z first.
00:28
The bounds for z will be the two planes first, zero, and then the other one will be pi over two minus x.
00:40
So there you have your z bounds.
00:43
Once you've integrated with respect to z, then you are now looking at a double integral in the xy plane, which it's always nice to have a just a little sketch.
00:56
So i'll draw it off to the side here, got a little nicer.
01:03
So with your xy plane, let's see what's left over.
01:08
Z is equal to zero, which means that x is equal to pi over two.
01:13
You'll get some vertical line segment here at pi over two.
01:19
And we also have the portion of the cylinder, y equals square root x, and then we have y equals zero.
01:29
So we get this region here, and i'm going to integrate with respect to y.
01:37
So i'll go from zero to square root x, and then finally the x bounds will go from zero to pi over two.
01:53
There we go.
01:55
So now we have y cosine of x plus z, and our order of integration, we have dz, dy, and dx.
02:11
So now it's all that's left is to evaluate the triple integral.
02:18
Okay, so let's go ahead and do this.
02:19
So i'm going to integrate with respect to z, so the outer two integrals remain intact.
02:34
All right, so we should have our constant y, and we should also have here a sine of x plus z.
02:42
We're evaluating z from zero to our pi over two minus x, and this will be dy dx.
03:06
All right, so let's see what happens here.
03:23
Okay, so if we plug in for z, this pi over two minus x becomes just sine of pi over two, which is one.
03:32
So we get y, and then if we plug in zero for z, we will have sine x, and i think it would be more appropriate if we factored out the y.
03:52
So we will have one minus sine, and then we will have...
04:11
There we go.
04:13
Great.
04:14
Now we're left with a double integral.
04:17
Let's integrate now with respect to y.
04:21
So we should get one half y squared...