Question

The area(in $\text{sq.units}$) enclosed between the region $x+y\le 6,x^2+y^2\le 6y$ and $y^2\le 8x$ is

• A. $\frac{5\pi }{2}-\frac{1}{2}$
• B. $\frac{9\pi }{4}-\frac{1}{6}$
• C. $\frac{7\pi }{4}+\frac{1}{3}$
• D. $\frac{9\pi }{2}+\frac{2}{3}$

Answer 1

The correct option is B $\frac{9\pi }{4}-\frac{1}{6}$

The region enclosed is shown below :
The required area $=$ Area of curvilinear triangle $OMRO$ + Area of trapezium $MNSR$ -Area of curvilinear triangle $ONSO$
$=\underset{0}{\overset{2}{\int }}\sqrt{8x}dx+\frac{1}{2}(4+3)\cdot 1-(3^2-\frac{\pi \cdot 3^2}{4})=\frac{9\pi }{4}-\frac{1}{6}$