Which of the following limits is equal to ?

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

Problem

Which of the following limits is equal to

\displaystyle\int_{1}^{4} 2x\,dx

? (a)

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\left(1+\frac{3i}{n}\right)\frac{3}{n}

(b)

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} 2\!\left(1+\frac{3i}{n}\right)\frac{3}{n}

(c)

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} 2\!\left(1+\frac{3i}{n}\right)\frac{1}{n}

(d)

\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} 2\!\left(1+\frac{i}{n}\right)\frac{3}{n}