Question

Use the limit process to find the area of the region between the graph of the function and the x-axis over the given y-interval. g(y) = 4y2 − y3, 1 ≤ y ≤ 3

Answer 1

the area is the limit of the sum of rectangles of height f(x) and width ∆x. That sum is (where ∆x = (3-1)/n = 2/n)
n
∑ f(n ∆x)*∆x = (4(k*2/n)^2 - (k*2/n)^3) * 2/n
k=1

= ∑32k^2/n^3 - 16k^3/n^4

now use your formulas for sums of powers of k to complete the evaluation.