00:02
All right, so our goal is to find the value of the integral from minus 5 to 5 of x minus square root of 25 minus x squared dx.
00:14
And we want to do this by geometry.
00:17
And what i'm going to do is split this into two different integrals because of that minus sign.
00:23
And we will do each of these integrals using geometry, using shapes.
00:31
Ok, so the idea is that it's very helpful if you know what these parent functions look like.
00:37
So let's do the first one.
00:39
So if i want to graph a y equals x, then we get a nice line here for y equals x.
00:49
And if we want to integrate from minus 5 to 5, that's the same as finding the area under the curve from minus 5 to 5.
00:57
But notice due to symmetry, on the right, this will be a positive area.
01:03
And on the left, this will count as a negative area.
01:06
And those cancel.
01:07
So due to symmetry, then we actually get a 0 for the first integral.
01:13
Now let's take a look at the second integral.
01:16
Ok, so let's do a quick graph here.
01:19
We're going to graph the second function.
01:22
But that is just the top half of a circle.
01:25
Because if i did x squared plus y squared equals 25, that would be a circle of radius 5...