Question

(5 pts) Set up, but do NOT evaluate, the integral needed to determine the mass M of a plate with a radius of 16 cm and a radial density given by rho(x)=1+2sin(x), in Kg/cm², with x in cm.

Answer 1

To find the mass of a plate with a radius of 16 cm and a radial density function rho(x)=1+2sin(x), we set up a double integral in polar coordinates over the entire plate area without evaluating the integral.

Step-by-step explanation:

To determine the mass M of a plate with a given radial density function, we need to set up an integral over the area of the plate. Because the density varies with the radius, we will use polar coordinates to express the area element dA and the variable x will correspond to the radial distance from the center.

The density function is given by ρ(x) = 1 + 2sin(x), where x is in cm and the plate has a radius of 16 cm. The mass can be found by integrating the density function over the plate's area, which can be represented as:

M = ∫∫_A ρ(x) dA

In polar coordinates, the area element dA is given by r dr dθ, with r being the radial distance and θ being the angular coordinate. Thus, the integral to find the mass M is:

M = ∫_0^{2π} ∫_0^{16} (1 + 2sin(r)) r dr dθ

Note that we integrate r from 0 to 16 and θ from 0 to 2π to cover the entire plate area.