00:01
So we need to decide if we want to integrate this in terms of x or y first.
00:05
In this case for the y, since we have y times sine x plus y, we're going to need to use integration by parts.
00:12
But if we integrate by x first, we can save that till the end.
00:15
So let's integrate by x first.
00:17
So we're going to have dx, d y, as the order.
00:20
And we have y, sine of x plus y is while we're integrating.
00:25
And we need to plug in our boundaries.
00:27
So x first and x goes from negative pi to zero.
00:30
So that's on the inside.
00:33
And y second goes from zero to pi.
00:37
So now we can integrate the inside and see what we did.
00:40
So we'll integrate this here.
00:44
And so this y is a constant.
00:45
So we're going to have y.
00:47
And the integral of sine of x plus y is negative cosine of x plus y.
00:53
So that's what we're going to have.
00:55
And we'll evaluate this from x equals negative pi to zero.
01:02
And so plugging in a zero for x, we have negative y cosine of zero plus y which is just y and then minus negative and so we're plugging in negative pi here so we have negative y cosine of negative pi plus y so we're just going to write y minus pi and these two negatives cancel so we're going to get looks like this so we have negative y cosine y plus y plus y cosine y plus y cosine y minus pi.
01:40
And so we have to integrate this in terms of y now using our outside integral.
01:45
So zero to pi and d y.
01:50
So for these y cosine y and y cosine y and y cosine y.
01:55
This should be a minus a row plus y minus pi.
02:00
We're going to have to use integration by parts.
02:02
So one thing we can do is pull out the y.
02:05
So we can have y times negative cosine y plus pose...