Triple Integration with spherical coordinates and Pappus centroid theory
Uncertain about an homework textbook question. It reads:
Sketch the solid region contained within the sphere x^2 + y^2 + z^2 = 16, and outside of the cone, z = 4 - sqrt(x^2 + y^2). Does anybody know of a free 3D graphing program/calculator online.
I can kind of picture it in my head, the region is basically above the xy plane, and thus, only the positive root of z = sqrt(16 - (x^2 + y^2)) needs to be taken into account. I also know that both the cone and the sphere intersect the z axis at (0,0,4) and this is also and intersection point for the curves.
Clearly identify the limits of integration using spherical coordinates and set up the iterated triple integral which gives the volume bounded by the above sketch. Do not evaluate the integral!
We have, sqrt(x^2 + y^2) = r = row(sin(phi)).
x^2 + y^2 + z^2 = 16 = row^2, therefore row = sqrt(16) = 4. We subsitute this into the above equation and let z = 0 to find phi:
z = 0 = 4 - row(sin(phi)) = 4 - 4sin(phi) --> phi = sin^-1(-4/-4) = pi/2.
So the limits of integration are:
Since there is a circle in the xy plane of x^2 + y^2 = 16, we can integrate with respect to theta from 0 to 2pi.
For phi, we determined above that the upper limit was pi/2, since the region is above the xy plane, the lower limit is 0.
We also determined that row had a value of 4 which is the upper limit and the lower limit is 0.
So the iterated triple integral excluding the limits is:
integral d(theta) integral(sin(phi)d(phi) integral((row^2)*(4 - row(sin(phi)))d(row). Right?!
Evaluate the volume given by the iterated triple integral above using pappus theory:
V(E) = 2pi*centre of mass y*A(D) = the volume of E is the area of D times the circumference o fthe circle traversed by the centroid (x,y) of D.
I am confused because Pappus theory is supposed to simplify volumes. However, in order to find the centroid, I need to find the centre of mass first and the moment of the lamina with respect to the x-axis.