Question

Let $f$ and $g$ be continuous function on $a\le x\le b$ and set $p\left(x\right)=$max$\{f\left(x\right),g\left(x\right)\}$ and $q\left(x\right)=$min$\{f\left(x\right),g\left(x\right)\}$, the area bounded by the curves $y=p\left(x\right),y=q\left(x\right)$ and the ordinates $x=a$ and $x=b$ is given by

• A. ${\int }_a^b(f\left(x\right)-g\left(x\right))dx$
• B. ${\int }_a^b(p\left(x\right)-q\left(x\right))dx$
• C. ${\int }_a^b|p\left(x\right)-q\left(x\right)|dx$
• D. ${\int }_a^b|f\left(x\right)-g\left(x\right)|dx$

Answer 1

Answer: (C), (D)

The correct options are
C ${\int }_a^b|f\left(x\right)-g\left(x\right)|dx$
D ${\int }_a^b|p\left(x\right)-q\left(x\right)|dx$

Here,curve $ABCD$ is max$\{f\left(x\right),g\left(x\right)\}=p\left(x\right)$ and curve $EBCF$ is min$\{f\left(x\right),g\left(x\right)\}=q\left(x\right)$
Thus, the area can be determine by either $\left(c\right)$ or $\left(d\right)$