# Question 10

## You MAY NOT use a calculator to solve this problem.

A flat metal plate one centimeter thick is shaped like the first quadrant region under the graph of between *x* = 0 and *x* = 4. The density of the plate varies; at a distance *x* units from the y-axis the density is given by x^{2} grams/cm^{3}.

(a) Assume the approximate mass, ΔM, of a strip of width Δx is:

ΔM = (volume)(density)
Write a left-hand Riemann sum with 4 terms that approximates the total mass of the metal plate.

(b) Write a definite integral that gives the exact value of the total mass of the plate.

(c) Determine the total mass of the plate.

**Solution;**

(a)

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