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 $A(x)=\int\limits_a^x f(x) d x \forall x \in[a, b]$, if $f(x)>0$
Statement II If $A(x)=\int\limits_a^b f(x) d x+\left|\int\limits_b^b f(x) d x\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)

$\int\limits_a^b f(x) d x$ 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 $\int\limits_a^x f(x) d x$ 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
$A(x)=\int\limits_a^x f(x) d x$
If a curve lies below $x$-axis we take its absolute value
$\therefore$ Statement II A $\left.(x)=\int\limits_a^b f(x) d x+\mid \int\limits_b^c f(x)\right) d x \mid$ is true. When $f(x)>0$ for $x \in[a, b]$ and $f(x)<0$ for $x \in[b, c]$.