00:01
In this problem, we are given this integral over some curve y dx plus x dy.
00:13
We are going to evaluate this integral from 0, 0 to 1, 1 on the xy plane for different curves given in lots of questions in this problem.
00:29
Okay, so let's get started with the first one, which is apparently number 13.
00:34
Okay, the first curve is y equal to x squared.
00:40
So we have dy equal to 2x dx.
00:45
Since we have this x as the integration parameter, we should also emphasize that x changes from 0 to 1.
00:55
So we write i equal to integral from 0 to 1.
01:00
Okay, we have x squared dx plus x times 2x dx.
01:09
We get from 0 to 1 3x squared dx, which becomes x cubed from 0 to 1.
01:22
It's not dx here and the result is 1.
01:27
Okay, now let's do the second curve.
01:31
This time we have y equal to x.
01:34
So dy equal to dx and again x changes from 0 to 1.
01:42
Then we have i equal to integral from 0 to 1.
01:51
Okay, we have x dx plus x times dx.
01:59
This time we have integral from 0 to 1 2x dx, which becomes x squared between 0 and 1.
02:08
So the result is 1.
02:14
Okay, next.
02:18
This time we have some straight line portions on the xy plane and it goes like this.
02:25
We start at the origin 0, 0.
02:34
We go up to 0, 1 and then we go to this final point 1, 1.
02:43
So i'm going to call this the first portion.
02:46
This is the second portion.
02:49
So we have i equal to the first portion plus integral over the second part...