The continuous function is defined on

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

Problem

The continuous function

g

is defined on

0 \le x \le 6

. Selected values of

g

are shown in the table below.

$x\;\;(\text{units})$	$0$	$2$	$4$	$6$
$g(x)$	$1$	$3$	$2$	$5$

Define the accumulation function

B(t)=\int_{0}^{t} g(x)\,dx, \quad 0 \le t \le 6

. (a) Using a trapezoidal sum based on the subintervals in the table, approximate

B(6)

. (b) If

g

is known to be concave upward on

[0,6]

, indicate whether the approximation in part (a) is an overestimate or an underestimate, and explain why. (c) Estimate

B(4)

using Simpson's Rule with two subintervals whose endpoints correspond to the table values. (d) Given that

g

is increasing on

[0,6]

, determine and justify whether

B(t)

is concave up, concave down, or neither on this interval.