00:01
For this, problem we are to find the area of the region between f of x that's equal to sine of 2x and g of x that's equal to cosine of x over the closed interval, negative pi over 2 to pi over 6.
00:12
Now the first thing we have to do is to find the intersection between the curves.
00:16
To determine the intersection, you want to set them equal to each other.
00:19
We have sine of 2x equal to cosine of x.
00:23
We have 2 sine x times cosine x equals cosine x and then we subtract both sides by cosine x we have 2 sine x times cosine x minus cosine x equal 0 we factor out cosine x we have cosine x times 2 sine x minus 1 equals 0 so cosine x equals 0 or we have 2 sine x minus 1 equals 0.
00:52
Now, cosine x equals 0 if x is equal to negative pi over 2, and 2 sine x minus 1, will give us sine x equals 1 half, and this is only true if x is equal to pi over 6.
01:12
And since the intersections are just the boundaries of the given interval, then we will only have one region.
01:19
Note that 1 x is 0, f of 0 is 0, and g of 0 is 1.
01:25
So this will be our graph...