00:01
All right, so in this question we're asked to solve this integral, the integral of this function over r.
00:08
So in this case we don't have to worry about the order of integration, we're integrating over a rectangle, right? so x does not depend on y, y does not depend on x, so this is the integral.
00:17
Let's choose the order first.
00:19
So there's two y's and only one x, so let's leave the y outside, and let's integrate first with respect to x, right? so y sine of x plus y, and actually, so this is respect to x, actually the y can just go out, just a number, does not depend on x, so just a number multiplying by the sine, so we leave it like this.
00:41
And now let's see what we get.
00:43
So the integral of sine is minus cosine of x plus y, and because we're integrating with respect to x, this is when x is minus pi, and when x is equal to zero, and this is dy.
00:58
So we're going have two integrals, but let me first write it down.
01:01
So this will be, let's say, minus y, then we have, when x is zero, we have cos of y, and when x is minus pi, we have minus cos of y minus pi dy.
01:17
Let's simplify this a little bit, so we know what happens with the cosine.
01:22
So the cosine of an angle is here, and the cosine of this angle minus pi would be here, so the cosines are symmetric...