A function is continuous on

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

Problem

A function

g

is continuous on

[0,1]

. Consider the non-uniform partition of

[0,1]

defined by

x_0=0

and

x_i=\dfrac{i^{2}}{N^{2}}

for

1\le i\le N

. For each subinterval

[x_{i-1},x_i]

, choose the point

c_i=\dfrac{i(i-1)}{N^{2}}

. (a) Show that the width of the

i

-th subinterval is

\Delta x_i = \dfrac{2i-1}{N^{2}}

. (b) Write a Riemann sum

S_N

associated with this partition and these sample points that approximates

\displaystyle \int_{0}^{1} g(x)\,dx

. (c) Write the definite integral

\displaystyle \int_{0}^{1} g(x)\,dx

as the limit of the sums

S_N

obtained in part (b). (d) Suppose

g(x)=x

. Evaluate the limit explicitly and compare your result with the value found by the Fundamental Theorem of Calculus.