00:01
Hi there, so for this problem, we are asked to find the area of the shaded region.
00:08
And we are given that the function f of x is equal to x to the 4, this minus 4 times x to the 3, and this plus 6 times x square.
00:34
And the function g of x is also given, and that function is 4 times x plus 15.
00:49
Now, to determine the area, what we need to do is the following integral.
00:55
So the area is the integral from a to b to the upper function, in this case, as you can see from the graph, is the function g of x, minus, the function f of g, f of x, sorry.
01:13
This integrator over x.
01:17
Now, we need to determine the values of the of a and b, but those are the values where these two functions crosses.
01:31
So as you can see from the figure, a is going to be equal to minus 1, and b is going to be equal to 3.
01:43
So that's what we need to substitute in here.
01:46
So finally, the integral that we need to solve is the integral from minus 1 to 3 of the function g, which is 4 times x plus 15.
02:01
This minus the function f of x, which is x to the 4, plus 4 times x to the 4, 3 minus 6 times x square.
02:13
And this integrated over x.
02:17
So now we just can simply start solving this.
02:23
So we have the first term in here is 4 times x.
02:29
So the integral of that will be 4 times x squared divided by 2.
02:35
This plus 15 times x.
02:40
This minus x to the 5 divided by 5 plus x to the 4 because the 4 cancels with the 4 that we have in there minus 6 divided by 3 so that will be 2 times x to the 3 and then this evaluated from from minus 1 to 3 so let me just simplify something here.
03:24
We know that 4 divided by 2 is just 2.
03:28
So we will have in here 2 times x square.
03:33
Now we just need to simply evaluate this.
03:37
So at minus 1, we will have 2 times minus 1 to the square.
03:45
This plus 15 times minus 1.
03:49
This minus minus 1 to the 5 divided by 5...