00:01
We're given a graph with the shaded region and the function y equals negative x times x minus 3, and we're supposed to compute the shaded region using theorem 1.
00:10
So as we know, theorem 1 says that to compute a shaded region with endpoints x equals a and x equals b, and the function is f of x, then this is equal to the antiderivative capital f of b, minus f of a.
00:36
So given this, we can plug in our own information into this.
00:40
So we know that our endpoints go from zero to three.
00:44
And we also know that our function is negative x times x minus 3.
00:52
However, we can simplify this to make it easier to take the antiderivative of.
00:57
So we can just make this negative x squared minus, or that will be plus 3x.
01:06
So now to calculate the antiderivative, we know that the antiderivative of negative x squared is negative 1 3rd x cubed.
01:22
And then for 3x, that's just going to be x squared 3 over 2 times x squared.
01:31
And now we can just put our end points at the side here.
01:35
So now we can use the second part of theorem 1.
01:39
And plug our numbers into that.
01:42
So now that we have the antiderivative, all we have to do is plug a and b into it...
