00:01
Suppose you have this region bounded by the function y that's equal to pi over two times the product of cosine x and sine of pi plus pi sine x and then the xx is now based on our graph here if we choose a vertical strip to represent the region we find that a is equal to the integral from negative pi to negative pi two since it's in between negative pi and zero.
00:31
And then the difference between the upper and the lower function.
00:36
So that's zero minus we have pi over two times cosine of x, sine of pi plus pi plus pi sine x and then dx.
00:49
Plus we have the integral from negative pi over two to zero of the upper minus the lower function that'll be pi over 2 cosine of x times sine of pi plus pi sine of x and then d x.
01:11
Now the antiderivative of pi over 2 cosine of x times sine of pi plus pi sine x d x can be taken by u substitution.
01:26
So let's say u equals pi plus pi sine x.
01:32
Then from here we get d u that's equal to pi cosine of x, meaning d u over pi equals cosine of x d x.
01:45
So then by substitution this is equal to the integral of pi over two times sine of u...