Question

Area of the region $\left\{\right(x,y)\in R^2:y\ge \sqrt{|x+3|},5y\le x+9\le 15\}$ is equal to

• A. $\frac{1}{6}$
• B. $\frac{4}{3}$
• C. $\frac{3}{2}$
• D. $\frac{5}{3}$

Answer 1

The correct option is C $\frac{3}{2}$


From the graph,
Area of the region $=$ Shaded area $=$Area of trapezium $ABCD-$ Area of $ABE-$ Area of $ECD$
$=\frac{1}{2}\times 3\times 5-\underset{-4}{\overset{-3}{\int }}\sqrt{-x-3}dx-\underset{-3}{\overset{1}{\int }}\sqrt{x+3}dx$
$=\frac{15}{2}+\frac{2}{3}{\left[\right(-x-3{)}^{\frac{3}{2}}]}_{-4}^{-3}-\frac{2}{3}{\left[\right(x+3{)}^{\frac{3}{2}}]}_{-3}^1$
$=\frac{15}{2}-\frac{2}{3}-\frac{16}{3}=\frac{3}{2}$