Calculus 4 Question

Question

For the solid of constant density k bounded by the surfaces z = x^2, y = 2z, z = 2, y = 0,

a. Show the region using a 3d grapher and suitable zoom settings.

b. Use iterated integrals to find the center of mass of the region. Give your answer as a coordinate triple (x, y, z) in exact form.

Answer 1

The x coordinate of the center of mass is zero by symmetry.

The results for the coordinates of the center of mass do not depend on D, so D can be taken as D=1 without loss of generality.

The y and z coordinates can be most easily found by replacing the original problems with an equivalent two dimensional problem in y, z

There is a base triangle in the y,z plane with vertexes (0,0) ; (0,2) ; (4,2)

On this triangle, we can define a density den = 2 sqrt(z)

Then mass (actually volume with D =1) and the forms for the y and z components of the center of mass

can be set up as two dimensional concatenated integrals .

The result for the center of mass is (0, 10/7 , 10/7)

It is worth notice that the centroid of the base triangle is (0, 4/3, 4/3)

So 10/7 is greater than 4/3 as expected.

Answer 2

https://www.desmos.com/3d/c9918c29d8

Above is a link to graph required in this problem.

Perhaps you will be able to get the center of mass integral from the limits shown in the graph.

Good luck.