Question

What is the area of the region bounded by X-axis, the curve $y=f\left(x\right)$ and the two ordinates $x=\frac{1}{2}$ and $x=1$?

• A. $\frac{5}{12}$ sq unit
• B. $\frac{5}{6}$ sq unit
• C. $\frac{7}{6}$ sq unit
• D. $2$ sq unit

Answer 1

The correct option is A $\frac{5}{12}$ sq unit

$f\left(x\right)=\left|x-1\right|+x^2,x\in R$ if $x\ge 1$
$\begin{array}{c}f\left(x\right)=x-1+x^2\\ ifx<1\\ f\left(x\right)=1-x+x^2\end{array}$.
${\int }_{1/2}^{1}f\left(x\right)dx=Area$ here $f\left(x\right)$ will be
$\begin{array}{c}{\int }_{1/2}^1\left(1+x^2-x\right)dx={\left[x+\frac{x^3}{3}-\frac{x^2}{2}\right]}_{1/2}^1\\ =\frac{1}{2}+\frac{1}{3}\left\{\frac{7}{8}\right\}-\frac{1}{2}\left\{1-\frac{1}{4}\right\}\\ =\frac{12+7-9}{24}\\ =\frac{10}{24}\\ =\frac{5}{12}\end{array}$