Let for . For a positive integer , let denote the approximation of obtained by using…

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

Problem

Let

p(t)=\dfrac{1}{1+t^{2}}

for

0\le t\le 5

. For a positive integer

n

, let

M_n

denote the approximation of

\displaystyle \int_{0}^{5} p(t)\,dt

obtained by using the midpoint of each of the

n

uniform subintervals of

[0,5]

as the sample point, and let

T_n

denote the approximation obtained by averaging the left-endpoint and right-endpoint values on every subinterval. Suppose

n=5

. (a) Compute

\Delta t

and list the 5 subintervals of the partition. (b) Calculate

M_5

to four decimal places. (c) Calculate

T_5

to four decimal places. (d) Using the results of parts (b) and (c), write an inequality that bounds

\displaystyle \int_{0}^{5} p(t)\,dt

between two numerical values. (e) Explain which of the two approximations,

M_5

T_5

, is the overestimate and which is the underestimate, citing appropriate features of

p

on the interval.