00:01
Okay, hello.
00:02
So here we have our function r of t is equal to t times ihat plus sine of t times j hat plus cosine of t times k hat.
00:09
We have t between 0 and 2.
00:11
Now the length of the curve here, we know is given by while length l is equal to the integral going from a to b of the magnitude of the derivative of r of t.
00:24
So we first, we find the derivative, our prime of t.
00:29
So r prime of t.
00:31
Just take the derivative here component by component, and we see the derivative is going to be equal to i plus cosine of t j hat minus sine of t times k hat.
00:47
Okay.
00:48
So then the magnitude of the derivative is equal to the square root of each component squared.
00:58
So the square root of 1 plus cosine square plus sine squared.
01:02
Well, actually, so again, 1 plus cosine squared t plus sine squared t.
01:10
But we know that cosine square plus sine squared is just 1.
01:13
Therefore, this is equal to the square root of 1 plus 1, which is the square root of 2.
01:18
So the main 2 is the square root of 2...