00:01
All right.
00:02
For this question, we are asked to find a line integral.
00:05
R of t is representing the parametrization of the curve c, and we're given that, and we're also given the bounds for t as well.
00:13
This is some sort of helix.
00:14
I believe it's probably revolving about the x axis.
00:19
No matter, what we need is to rewrite our line integral as a single integral.
00:25
To do that, we're going to need the derivative of our parametization, which we can take component -wise.
00:39
And what we're also going to need, because this is a scalar function, we're going to need the magnitude of r prime.
00:48
And this magnitude is found using pythagorean theorem, sum of squares.
00:57
And we love seeing these trig functions because when you pull out the nine from each of those trig functions, you'll get sine squared plus cosine squared, which equals 1.
01:10
So you end up just with this root 1 plus 9, and you get square root 10.
01:17
Fantastic.
01:18
So now our line integral, the way this is going to work, is it will now become a single integral from the bounds of t that are provided, so from 0 to pi.
01:29
And then we have our evaluation, the function that's given, we evaluate with our parameter.
01:36
So we have y times z.
01:38
So in that case, we have 3 cosine times 3 .5.
01:51
Times cosine of x, so that would be cosine of t.
01:56
And to top it all off, we need to multiply by the magnitude of r prime, which is this constant square root 10.
02:04
And now this will be in terms of dt.
02:09
All right, so this is our single integral to solve.
02:11
Let's pull out all of the constants.
02:13
So i have two factors of three, and i have a square root 10.
02:20
And we can combine some trig functions together.
02:22
We can write cosine squared and sign.
02:35
This is where i would do u substitution...