Question

The area of the region bounded by the line $y=3x+2$, the $X$-axis and the ordinates $x=-1$ and $x=1$ is

• A. $\frac{26}{3}$ sq units
• B. $\frac{19}{3}$ sq units
• C. $\frac{13}{3}$ sq units
• D. None of the above

Answer 1

Answer: (C)

As shown in the figure, the line $y=3x+2$ meets $x$-axis at $x=\frac{-2}{3}$ and its graph lies below $X$-axis for $x\in \left(-1,\frac{-2}{3}\right)$ and above $X$-axis for $x\in \left(\frac{-2}{3},1\right)$.

The required area $=$ Area of the region $ACBA$
+ Area of the region ADEA
$=\left|\underset{-1}{\overset{\frac{-2}{3}}{\int }}(3x+2)dx\right|+\underset{\frac{-2}{3}}{\overset{1}{\int }}(3x+2)dx$
$=\left|{\left[\frac{3x^2}{2}+2x\right]}_{-1}^{\frac{-2}{3}}\right|+{\left[\frac{3x^2}{2}+2x\right]}_{\frac{-2}{3}}^1$
$=\frac{1}{6}+\frac{25}{6}=\frac{13}{3}$ sq units