Calculus Question

Question

For the solid of constant density k bounded by the surfaces z = x^2, y = 2z, z = 2, y = 0, 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. Show all your work.

Answer 1

In case the audio doesn't work, here's what I'm saying in the video:

The equation for mass comes from the formula for density, ρ = m/V. For oddly shaped solids or solids of varying density, it is more sensical to think of density for a small volume dV, so ρ(x, y, z) = dm/dV. Multiplying both sides by dV and adding up all the masses dm for the whole solid, you get the total mass of the solid, which is where the volume integral for m in the left column comes from.

For finding the center of mass, you use a similar integral. This time, instead of adding up all the dm, you add up all the dm times their position within the solid to get the moment, then you divide by the total mass to get the center of mass coordinate in question. These coordinates are denoted as x, y, or z with a bar on top (read "x-bar", "y-bar", and "z-bar", respectively).

The main things I left out were the limits of integration for x, y, and z. They will be slightly different depending on which order you integrate the variables in, and the surfaces bounding the surface that the question gives should help you figure these out.

Let me know if you get stuck, and I'll try to nudge you in the right direction.

-Zach