A continuous function is defined on by • for (the upper semicircle of radius centered at )

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

Problem

A continuous function

f

is defined on

-2\le x\le 6

by •

f(x)=\sqrt{4-(x+0)^2}

for

-2\le x\le 2

(the upper semicircle of radius

2

centered at

(0,0)

), and •

f(x)=-3

for

2\le x\le 6

. Let

g(x)=\int_{-2}^{x} f(t)\,dt

with

g(-2)=0

. (a) Find the exact value of

g(2)

. Show your work. (b) Determine the

x

–value in the interval

2\le x\le 6

at which

g

attains its minimum. Justify your answer. (c) Using your results, compute the average (mean) value of

f

on the entire interval

-2\le x\le 6

. Express your answer exactly.