00:01
Alright, in your question, you're asked to find the area of the shaded region.
00:04
Now, your question was not submitted with the picture, but your question references that this graph would have already been drawn because it says find the area of the shaded region.
00:15
If you graph these two equations, you end up creating two regions.
00:20
So what i'm going to do is i'm going to calculate the area of both regions, and i'm also going to calculate the area of those two regions combined.
00:28
Combined.
00:28
That way you can match it up with the graph you've been given.
00:32
You may have been given a graph where only region 1 was shaded, or you may be given one where only region 2 is shaded.
00:40
Or if you were given both, i'll have that answer as well.
00:43
The first thing we have to establish is the x -coordinates where the intersections of the two graphs lie.
00:51
So what i'm going to do is i'm going to take the first equation and i'm going to plug it into the second equation and solve for x.
01:03
I'll do that by adding x squared to both sides of the equation and subtracting 6x from both sides of the equation.
01:13
That gives you x cubed minus 16x equals 0 because the x squared is cancelled out.
01:22
We solve that by factoring out an x, giving us x squared minus 16 equals 0.
01:31
Continue factoring, we get x minus 4 and x plus 4 equals 0, which means that x equals 0 is an intersection, x equals 4 is an intersection, and x equals negative 4 is an intersection.
01:48
The x coordinate at this location would be negative 4, here here is x equals 0, and the one to the right is x equals 4.
02:00
Starting with region 1, we would integrate the higher function minus the lower function.
02:07
The higher function in that region, the red graph, was the first graph.
02:14
The second line you see is the parabola, the blue graph.
02:19
So i want to take the red graph, which is x cubed minus x squared minus 10x, and i want to subtract the blue graph, which is negative x squared plus 6x.
02:39
That's what we're doing on the first region from negative 4 to 0...