Integrate the function ((x^2+y^2)^frac14) over the region E...

Integrate the function ((x^2+y^2)^{frac{1}{4}}) over the region E that is bounded by the xy plane below and above by the paraboloid 10 - 9x^2 - 9y^2 using cylindrical coordinates. ???_E (x^2 + y^2)^{frac{1}{4}} dV = ?_A^B ?_C^D ?_E^F G(z, r, ?) dz dr d? where A = 0, B = 2pi, C = 0, D = sqrt(10)/3, E = 0, F = 10-9r^2 and G(z, r, ?) = (r^2)^(1/4)r. The value of the integral is ???_E (x^2 + y^2)^{frac{1}{4}} dV =

Transcript

00:01 In this problem, we are going to compute a triple integration in cylindrical coordinates.

00:06 So the integral is given by x squared plus y squared to the power one -third dv over a region, over this three -dimensional region e, that is bounded by the xy -plane, so it is above the, so we are on the, we are above the xy -plane, and it is below this profile function z equal to nine minus two times x squared plus y squared.

00:45 Okay, let us try to see what we are dealing with here.

00:48 So i have plotted this region using mathematica beforehand, so let me just paste my result.

00:55 So we are in, within this cap, above the orange plane and under this blue dome, kind of.

01:02 Okay, now we start with the volume element, dv equal to dx, dy, dz, and we need to work in cylindrical coordinates, so we need the transformation from the cartesian to cylindrical coordinates, and we already know these expressions.

01:26 Next, we need the jacobian of the transformation to write down the new volume element in terms of r, theta, and z.

01:33 So it is given by a determinant of this matrix, partial x over partial r, partial x over partial theta, partial x over partial z, and we have the same column, but with y instead of x, and we have the same column with z instead of x.

01:54 So we obtain the well -known result, j equal to r.

01:59 Okay, now we write dv equal to j, dr, d theta, dz, or r, dr, d theta, dz.

02:09 Then we need to decide on the limits of integration, so we need to go back to the figure.

02:16 So this is the z -axis, and we are going all the way around the z -axis.

02:21 Therefore, theta should be integrated from 0 to 2 pi.

02:31 Next, we have r changing from 0 to this maximum radius.

02:37 This maximum radius is given by this equation with z equal to 0, because z is the defining circle for this intersection.

02:48 If we do that, we have x squared plus y squared equal to 9 over 2, this is r squared, so r is equal to 3 over root 2.

02:59 This is the maximum radius, so we should integrate r from 0 to 3 over root 2.

03:08 And finally, we have z.

03:11 Let's plot it like this.

03:14 We have this cap, okay, not here, this is r, this is z.

03:19 We have this cap, and this profile changes with respect to r, with respect to this radial distance.

03:27 So for any force, this arbitrary value of r, z changes from 0 to this profile function, which is 9 minus 2 times x squared plus y squared, or 2 times r squared.

03:42 So this is the limit of z minus 2r squared.

03:51 And then we are ready to write down everything, this integral in an open form.

03:59 So i equal to dv f of x, not x, y, z, r theta z.

04:06 So we have r dr d theta dz f of r theta z.

04:16 Now we need to break down this integral, this volume element, so as to show the limits...

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