Which of the following limits equals the definite integral after the algebra in the…

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Which of the following limits equals the definite integral ∫02(x2+4x) dx\displaystyle \int_{0}^{2}(x^{2}+4x)\,dx after the algebra in the summand has been fully simplified? (a) lim⁡n→∞∑i=1n(4i2n3+8in2)\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{4i^{2}}{n^{3}}+\frac{8i}{n^{2}}\right) (b) lim⁡n→∞∑i=1n(2i2n3+8in2)\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{2i^{2}}{n^{3}}+\frac{8i}{n^{2}}\right) (c) lim⁡n→∞∑i=1n(8i2n3+16in2)\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{8i^{2}}{n^{3}}+\frac{16i}{n^{2}}\right) (d) lim⁡n→∞∑i=1n(8i2n3+8in2)\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{8i^{2}}{n^{3}}+\frac{8i}{n^{2}}\right)

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