Consider a plate with a radius of 8 and a radial density given by ρ(x)

Question

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Consider a plate with a radius of 8 and a radial density given by ρ(x) = 4x^2/3 + 5. What is the mass of the plate? (Enter answer using exact value.)

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Consider a plate with a radius of 8 and a radial density given by ρ(x) = 4x^2/3 + 5. What is the mass of the plate? (Enter answer using exact value.)

Explanation & Steps

A circular plate with radius r and radial density function ρ(x) is given in the question. The radial position x within the plate determines the radial density. The radial density function describes how the density changes as x fluctuates. Integrate the product of the density function and the circumference of each circular ring inside the plate to determine mass m. Set up the integral by multiplying the radial position (x) by the density function ρ(x) and the circumference of the circular ring (2πx).