00:01
In this question, we have to use the divergence theorem to calculate the flux integral, flux integral, which is the surface integral integral integral fds.
00:11
We have to find out this and double integral fds.
00:16
We have to find out the value of this where f is given to be x -rays to the power 4i minus x -cube z -square j plus 4x y -square z -k.
00:24
And given it is bounded by the surface of solid bounded by the cylinder, x square plus y square equals to 9 z equals to x plus 4 and z equals to 0 now first of all using the divergence theorem we will first of all find the divergence of f which is dale f which will be equivalent to partial differentiation of x raised to the power 4 with respect to x plus partial differentiation with x y of minus x cube z square and partial differentiation with respect to z of 4x y square z now differentiating all of these we will have divergence of f equivalent to 4x q plus 0 since it does not have any term which is any term in terms of y plus partial differentiation with respect to z will give us 4 x y square so finally we will have divergence of f is equivalent to 4x x square plus y square now by by by by by by by by by by by by by by by by by by divergence theorem by divergence theorem, the flux will be equivalent to double integral, which is double integral sf .ds, will be equivalent to triple integral divergence f dv over the vchon, where v is the vgen.
01:53
Now, substituting the values, we will have triple integral value of divergence we have just calculated, 4x multiplied by x squared plus by, square db.
02:03
Now we will use cylindrical coordinates.
02:08
We are given that the surface is bounded by the cylinder x square plus y square equals to 9 and the surface z equals to x square plus 4 which will be something like this and the plane z equals to 0.
02:21
So this will be our surface.
02:24
Now using the cylindrical coordinates by cylindrical coordinates the volume bounded by this surface is going to be first of all, we will substitute x equals to r cost theta, y equals to r sine theta, and z will remain z...