What is the correct answer

Question

Which of the following limits is equal to ∫₃ ⁵ x⁴ dx?

A. lim _x→∞ ∑ _k=1 ⁿ (3 + k/n)⁴ ⋅ 1/n

B. lim _n→∞ Σ_k=1 ⁿ (3 + k/n)⁴ ⋅ 2/n

C. lim _n→∞ Σ_k=1 ⁿ (3 + 2k/n)⁴ ⋅ 1/n

D. lim _n→∞ Σ_k=1 ⁿ (3 + 2k/n)⁴ ⋅ 2/n

Answer 1

We are summing the area of n rectangles from 3 to 5. Thus, the base of each rectangle is 2/n. The functions are evaluated at x incremented by 2/n from 3 to 5, so it will be evaluated at 3 + 2k/n for k = 1, ...n

Thus, our answer is D.