00:01
R is the region bounded by f of x equals negative x squared over 3 plus 4x over 3 minus 10 over 3 and g of x is x squared over 3 minus 10x over 3 plus 10x over 3.
00:12
So r is the region bounded by those functions.
00:15
We want to find the area of r.
00:18
So we first need to figure out where those functions intersect.
00:22
Where does f of x equal g of x? so that would be negative x squared over 3 plus 4x over 3 minus 10 over 3.
00:34
We set that equal to x squared over 3 minus 10x over 3 plus 10 over 3.
00:42
So we could first multiply both sides by 3 to cancel out all those fractions.
00:48
That will leave us with negative x squared plus 4x minus 10 equals x squared minus 10x plus 10.
00:56
And if we, let's see, we could subtract x squared from both sides, add a 10x to both sides and subtract a 10 from both sides.
01:12
That gives us negative 2x squared plus 14x minus 20 and that's now equal to zero.
01:20
We factor out a negative 2 and that leaves us with x squared minus 7x plus 10 and that's equal to zero.
01:28
And we can factor that quadratic trinomial into x.
01:35
See x squared minus 7x plus 10, that would factor into x minus 2 times x minus 5.
01:43
And we have our factors of x minus 2, that means x equals 2 would be a solution and the factor of x minus 5 means x equals 5 would be a solution.
01:53
So those are the two times the functions intersect.
01:55
So roughly a graph of this might look like this.
02:00
We might have our f of x function which is a parabola that opens downward.
02:08
It might look roughly like that.
02:11
So there's our f of x.
02:13
Then our g of x function is a parabola that opens upward and it might look something like that.
02:22
We know the two x values, they intersect...