Consider the definite integral

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

\displaystyle \int_{1}^{4} \bigl(3x^{2}+2x\bigr)\,dx

. The interval

[1,4]

is divided into

N

congruent subintervals, and the midpoint of each subinterval is chosen as the sample point. (a) Derive a formula for the width

\Delta x

of each subinterval and for the

i

-th midpoint

x_i^*

in terms of

N

and

i

. (b) Using the expressions from part (a), write an explicit limit of Riemann sums that is equal to

\displaystyle \int_{1}^{4} \bigl(3x^{2}+2x\bigr)\,dx

. Your expression must involve

\displaystyle \lim_{N\to\infty}

and a single sigma notation summation. (c) Evaluate the limit obtained in part (b) to verify the numerical value of the integral.