Let be a piecewise-defined function with the following description

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

Problem

Let

r

be a piecewise-defined function with the following description. • On

[0,2]

, the graph of

r

is the line segment joining

(0,0)

and

(2,3)

. • On

[2,4]

, the graph of

r

is the upper semicircle of radius

1

centered at

(3,0)

. • On

[4,6]

, the graph of

r

is the line segment joining

(4,0)

and

(6,-2)

. Another function is given by

s(x)=2x-5

for

0\le x\le 6

. (a) Evaluate

\displaystyle \int_{0}^{6} r(x)\,dx.

Show all geometric reasoning used to obtain exact values. (b) Determine

\displaystyle \int_{0}^{6} s(x)\,dx.

r

s

, has the greater net signed area over

[0,6]

, and justify your answer using your calculated integrals.