00:01
Okay, we want to find the area enclosed by these four equations.
00:05
So we're going to have to draw it to see what it looks like.
00:09
The first one is nine times one over x.
00:13
So we know that 1 over x, well, you know that 1 over x looks like this.
00:19
So what's the 9 do? well, let's put some numbers in.
00:23
When x is 1, y is 9.
00:27
And when x is 9, y is 1.
00:30
And when x is 3 y is 3 all right so it looks sort of like this oops try one more time okay i'm gonna try one more time all right and then y equals x that's the line that goes through zero zero at 45 degrees and then y equals zero that's the x axis and x equals nine that's right here all right so the area enclosed is this little space right here okay so that's what we're going to try and find all right so first we need to decide should we cut this vertically or should we cut it horizontally okay let me let me fix this curve a little bit more okay because this needs to be curvy like that all right so if i cut it vertically some of the pieces have the line y equals x on the top and some of the pieces have the hyperbola y equals 9 over x okay so that's not my favorite okay if we cut it horizontally okay some of them have the red as the left and the blue as the right oh some of them have red on the left and the black on the right.
02:41
I'm going to cut it the first way.
02:44
All right.
02:44
I'm going to cut vertically.
02:46
So i'm going to have to do two integrals, one that has y equals x on the top and one that has y equals 9 over x on the top.
02:58
So i need to know what is this point right here where they cross because that's when it's going to stop being the first integral and start being the second integral.
03:11
Well, it's where y equals 9 over x, the black curve intersects y equals x.
03:23
So i'm just going to set them equal to each other.
03:26
I get the y out of there.
03:28
9 over x equals x, cross multiply, 9 equals 9 equals x squared.
03:34
So x equals plus or minus 3, but we're only interested in the positive one.
03:41
All right...