00:01
Right, this problem we need to find area of the shaded region.
00:06
So by formula, we know if we want to integrate in spade of x, first we write down the integral.
00:15
So the area a equals to integral in terms of x.
00:25
And we know the shaded area, x goes from 0 to 1.
00:29
So we can write the bounded down from 0 to 1.
00:34
Okay, and for here, the formula is the upper curve minus the lower curve.
00:44
So this will be e to the power x minus x times e to the power x square.
00:55
So our next step basically is try to evaluate this into the all right so we can do it separately the first term will be integral from 0 to 1 e to the power x d x minus integral from 0 to 1 x x times e to the power x okay and the first that we know the antiderivative of that this is e to the power x so this is such as e to the power x evaluate x equals 1 minus x equal to 0.
01:51
And this term here, we can bring this x to the differential that will give us x square.
02:00
So before we do that, we need to times 2.
02:03
So the trick we are using here, we times 2 and it divided by 2.
02:10
So the second term will be 1 half times integral 0 to 1, e to the power x square, d x square.
02:28
And as we can see this, if we use change of variable, say this x square to be another variable t, then this is e to the power t...