Question

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 p(x)=1+2sin(x) within the interval a,b.

Answer 1

To determine the mass M of a plate, consider concentric cylindrical shells of thickness dr and set up the integral over the radius of the plate with the provided density function. Therefore, the integral to set up (but not evaluate) for the mass M of the plate is: ∫016 (1 + 2sin(r)) 2πr dr.

Step-by-step explanation:

To determine the mass M of a plate with a given radial density, we need to set up an integral that accounts for the continuously varying density. To do this, we can consider concentric cylindrical shells within the plate, each shell having a tiny thickness dr. Since the density function is given as p(x) = 1 + 2sin(x), the elemental mass dm for each shell would be the product of the density at that radius times the surface area of the shell, which is 2πr dr (where r is the radial distance from the center). The limits of the integral will be from 0 to the radius of the plate, which is 16 cm in our case. Therefore, the integral to set up (but not evaluate) for the mass M of the plate is:

∫016 (1 + 2sin(r)) 2πr dr

This integral, when evaluated, would give us the total mass of the plate.