00:01
In this question, we have to evaluate the integral x plus z t s over the region where s is a part of the cylinder y square plus z square equals to 4 and x belongs to 0 to 8.
00:14
So if i'll try to make a graph for it, so we have a cylinder x y square plus z square equals to 4.
00:24
So it will be something like this.
00:27
If this is x -axis, this is y and this is z, it will be a cylinder, something.
00:32
Like this.
00:33
Now since we have x equals to 0, suppose 8 is here and we need x equals to 8.
00:40
So it will be a cylinder like this.
00:42
So this is the cylinder that we need.
00:44
This is the surface.
00:46
Now to solve this we will use parametric equation.
00:49
Let since the radius is 2, we can let x equals to v, y equals to 2 cause you since the radius is 2 and z equals to 2 sine u.
01:03
Okay, then we will have coordinates as ruv equals to v comma 2 cos u, comma 2 sine u.
01:12
Now first of all we will try to find the value of t s for this we need ru which is the partial differentiation with respect to you this will give us 0 comma minus 2 sine u and partial differentiation of 2 sine u will give us 2 cos you...