00:01
So in this question, we're going to start by finding the area under the curve, y equals the cosine of x, over the interval from 0 to pi over 2.
00:08
So to find the area under this curve, we need to compute a definite integral.
00:15
The definite integral of the cosine of x, dx, on the interval that i've been given, which is from 0 to pi over 2.
00:27
So how do i evaluate this definite integral? i start by finding an antiderivative.
00:34
What's the antiderivative of the cosine of x? the antiderivative of the cosine of x is the sign of x.
00:44
And we are going to evaluate that on the interval from 0 to pi over 2.
00:53
I plug in my top limit of integration, pi over 2, and i subtract from that what i get when i plug in my bottom line, limit of integration? zero.
01:09
What's the sign of pi over two? sign of pi over two is one, minus what's the sign of zero? sign of zero is zero, giving us one minus zero, which is one as our answer.
01:30
Now in b, they say we want to make a conjecture about the integral from zero to pi of the cosine of x -d -x, and confirm that conjecture using the fundamental theorem of calculus.
01:45
So from 0 to pi, my suspicion is that the value of this integral is 0...