00:01
We want to find the area of the enclosed region.
00:04
We have the equations, y equals 1 quarter x squared.
00:11
That's the wide parabola.
00:12
We have 2x squared, which is the narrow parabola.
00:16
We have x plus y equals 3, which is the equation of a line.
00:22
So let's label that x plus y equals 3.
00:26
And we're only interested in x greater than 0.
00:30
So we need to figure out what our limits of integration will be.
00:32
We have this first point, we have this second point, and this third point.
00:36
Let's label them a, b, and c.
00:40
A is the intersection of our two parabolas.
00:44
2x squared equals 1 quarter x squared.
00:48
So this tells us that x is equal to 0.
00:52
That's the origin.
00:54
Looking at the second point of intersection, we have to solve for 2x squared, our narrow parabola, equals the equation of our line.
01:06
So our line is y equals 3 minus x.
01:10
So this is just a quadratic.
01:13
So solving it gives us two solutions.
01:16
We have one of them is x equals 1 and the other is x equals negative 3 over 2.
01:24
But we only care about x being positive.
01:26
So we take x equals 1.
01:29
Next we're looking at the third point of intersection, which is going to be the intersection of our wide parabola, 1 quarter x squared, and our line, y equals 3 minus x.
01:42
Again, it's a quadratic, so we can solve it by using the quadratic formula or by factoring.
01:48
And what we get is we have the solutions x equals negative 6 or x equals 2, but since x is positive, we take x equals 2.
01:58
So we can label these points along the x -axis.
02:02
This is x equals 0, x equals 1.
02:07
And x equals 2...