Question

Area of the region bounded by the curve $y=|x+1|+1$, $x=-3,x=3$ and $y=0$ is

• A. 8 sq units
• B. 16 sq units
• C. 32 sq units
• D. None of these

Answer 1

Answer: (B)

Given equation of the curves are
$y=|x+1|+1=\{\begin{array}{l}(x+1)+1,\text{ if }x+1\ge 0\\ -(x+1)+1,\text{ if }x+1<0\end{array}$
$=\{\begin{array}{llc}x+2,& \text{ if }& x\ge -1\\ -x.& \text{ if }& x<-1\end{array}\dots .(1)$
$x=-3...(ii)$
$x=3...(iii)$
and $y=0...(iv)$
eq. (ii) represents the line parallel to y-axis and passes through the point $(-3,0)$.

eq. (iii) represents the line parallel to y-axis and passes through the point $(3,0)$
$\therefore$ Required area $=\underset{-3}{\overset{-1}{\int }}ydx+\underset{-1}{\overset{3}{\int }}ydx$
$=\underset{-3}{\overset{-1}{\int }}-xdx+\underset{-1}{\overset{3}{\int }}(x+2)dx$
$={\left[\frac{-x^2}{2}\right]}_{-3}^{-1}+{\left[\frac{x^2}{2}+2x\right]}_{-1}^3=16$ sq. units