Let on the interval

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

Problem

Let

f(x)=\sqrt{x}

on the interval

[0,1]

. For a positive integer

N

, divide

[0,1]

into

N

equal subintervals and choose the right endpoint of each subinterval as the sample point. (a) Express the width

\Delta x

of each subinterval and the

i

-th right endpoint

x_i

in terms of

N

and

i

. (b) Determine an explicit limit involving

N

\sum

, and

f(x_i)

whose value equals

\displaystyle \int_{0}^{1}\sqrt{x}\,dx

. (c) Compute the limit from part (b) to obtain the exact value of the integral.