Two functions and are defined on the interval as follows

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Two functions

f

and

g

are defined on the interval

0\le x\le 6

as follows. • For

0\le x\le 3

f(x)=3

and

g(x)=x

. • For

3\le x\le 6

f(x)=6-x

and

g(x)=0

. The graphs of

f

and

g

bound several regions in the plane. All questions below must be answered without evaluating any antiderivatives. (a) On the axes you create, sketch the graphs of

y=f(x)

and

y=g(x)

for

0\le x\le 6

, clearly labeling all intercepts and intersection points. (b) Compute the exact area of the region where

f(x)\ge g(x)

0\le x\le 3

. Explain how geometric formulas support your calculation. (c) Compute the exact area of the region where

g(x)\ge f(x)

3\le x\le 6

. Justify your work geometrically. (d) Using your answers to parts (b) and (c), determine the numerical value of

\displaystyle\int_{0}^{6}\bigl(f(x)-g(x)\bigr)\,dx

. Interpret the sign of this integral in the context of the regions you found.

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