00:01
In this problem, we are given a three -dimensional vector function f.
00:06
We want to use the divergence theorem to find the surface integral f.
00:11
N -hat d 's.
00:12
Or s here corresponds to a surface that is the boundary of the solid between z is equal to 0 and z is equal to x square.
00:21
Or x ranges between 0 and 2 and y ranges between minus 2 and 2.
00:28
So the first thing we're going to want to do is to state the divergence theorem.
00:41
According to the divergence theorem, a surface integral over a vector field can rewrite to a volume integral of the divergence of our vector field.
01:03
So we want to evaluate this last integral here, and to do so, we're going to need to evaluate our integrat, the divergence of f.
01:15
So generally speaking, the divergence of a three -dimensional vector field is equal to the partial derivative with respect to x of fx the x component or a vector field plus the partial derivative with rectal y of f y plus the partial derivative with the vector z of f c so in our case f x is equal to x squared f y is equal to zero and f z is equal to x times c play by in this expression we will obtain that the divergence of f or our given vector field is equal to 2x plus x, reducing to 3x.
02:19
Which means that our surface integral will rewrite as the volume integral of 3x dv.
02:28
Where dv and cartesian coordinates we write to the x times the y times d z.
02:37
So now the only thing left to do to evaluate this volume integral is to adequately determine our integration limits.
02:49
So in our problem statement we are told that x will range between 0 and 2.
02:57
Y will range between minus 2 and 2 and z will range between 0 and d and x square.
03:12
The only problem now is that we will have a relation for x and z.
03:18
The problem is we're integrating with respect to x first but our limits on z will also depend on x.
03:24
So we want to integrate with respect to x first.
03:29
So we want that our limit here on x to depend on z...