Question

Consider the following statements
Statement I $y=f(x)$ is a curve and $A(x)$ is the area bounded by the curve $f(x),x=a$ and $x=b$, then $A(x)=\underset{a}{\overset{x}{\int }}f(x)dx\forall x\in [a,b]$, if $f(x)>0$
Statement II If $A(x)=\underset{a}{\overset{b}{\int }}f(x)dx+\left|\underset{b}{\overset{b}{\int }}f(x)dx\right|$ if $f(x)>0$ for $x\in [a,b]$ and $(x)<0$ for $x\in [b,c]$ Choose the correct option.

• A. Statement I is true
• B. Statement II is true
• C. Both statements are true
• D. Both statements are false

Answer 1

Answer: (B)

$\underset{a}{\overset{b}{\int }}f(x)dx$ is the area of the region bounded by the curve $y=f(x)$, the ordinates $x=a$ and $x=b$ and $x$-axis. Let $x$ be a given point in $[a,b]$. Then $\underset{a}{\overset{x}{\int }}f(x)dx$ represents the area of the light shaded region in figure [Here it is assumed that $f(x)>0$ for $x\in [a,b]$, the assertion made below is equally true for other functions as well] . The area of this shaded region depends upon the value of $x$.

In other words, the area of this shaded region is a function of $x$. We denote this function of $x$ by $A(x)$. We call the function $A(x)$ as area function and is given by
$A(x)=\underset{a}{\overset{x}{\int }}f(x)dx$
If a curve lies below $x$-axis we take its absolute value
$\therefore$ Statement II A $(x)=\underset{a}{\overset{b}{\int }}f(x)dx+\mid \underset{b}{\overset{c}{\int }}f(x))dx\mid$ is true. When $f(x)>0$ for $x\in [a,b]$ and $f(x)<0$ for $x\in [b,c]$.