Question

Find the mass of the solid bounded by the planes x = 0, x = 2, y = 0, y = 3, and the surface z = x² + y².

Answer 1

To find the mass of the solid bounded by the given planes and surface, calculate the triple integral of the density function over the given region.

Step-by-step explanation:

To find the mass of the solid bounded by the given planes and surface, we need to calculate the triple integral of the density function over the given region. The density function is given by ρ = 1.

We can express the given region as R: 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ x² + y².

The mass M of the solid is given by the triple integral of the density function ρ over the region R:

M = ∭ ρ dV = ∭ 1 dV

Using cylindrical coordinates, the integral can be simplified as follows:

M = ∭ 1 dV = ∬ r dr dz dθ

Now we can evaluate the integral:

M = ∫02π ∫02 ∫0r^2 r dz dr dθ

Calculating this integral will give us the mass of the solid bounded by the given planes and surface.