00:01
This problem has to evaluate this integral, and i will do that by using trigonometric substitution.
00:07
So first i'm going to let y equal 1 half times secant of theta.
00:16
And the reason why you would choose that is because of what's within this square root.
00:22
The fact that it's a variable squared minus a constant.
00:26
So let me calculate what dy is.
00:29
This is 1 half times secant of theta times tangent of theta.
00:38
And one other thing, we have bounds on theta.
00:42
It's either in the first quadrant or the third quadrant.
00:48
So this is one of those things where you can just choose to memorize it, or you can sort of figure it out for yourself.
00:52
So i'll have a little bit more to say on that once we rewrite this integral.
00:57
So the last thing is to find the new limits of integration.
01:02
We see that y is equal to negative 1, but y is also equal to 1 half times secant of theta.
01:10
So i could write this as 1 half times 1 over cosine of theta.
01:18
And that means that cosine of theta, this is negative 1 half.
01:26
So theta is not in the first quadrant.
01:27
It has to be in the third quadrant.
01:31
So this implies that theta is equal to 4 pi over 3.
01:38
Then for the next limit of integration, we have that negative square root of 2 over 2.
01:48
This is equal to, again, 1 half times 1 over cosine of theta.
01:56
2's cancel.
01:58
We have that cosine of theta.
02:01
This is negative 1 over square root of 2, which is also equal to negative square root of 2 over 2.
02:12
Again, cosine is negative.
02:14
So that means that it has to be in the third quadrant with this substitution that we've made.
02:20
So theta is equal to 5 pi over 4.
02:27
And now after all of that, we can finally rewrite this integral.
02:31
So this is equal to integral from 4 pi over 3.
02:38
This is 5 pi over 4.
02:42
And then dy, we calculated that here.
02:46
This is 1 half secant of theta times tangent theta.
02:58
And then i have y, which is 1 half secant of theta.
03:07
And then times, this will be 4 times 1 fourth secant squared minus 1...