Question

The area (in sq. units) of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is:

• A. $\frac{3}{4}$
• B. $\frac{9}{8}$
• C. $\frac{5}{4}$
• D. $\frac{7}{8}$

Answer 1

The correct option is B $\frac{9}{8}$

$x^2=4y$ is a parabola with vertex $(0,0)$ and focus $(0,1).$
$x=4y-2$ is a straight line with intercepts $(-2,0)$ and $(0,\frac{1}{2}).$
Points of intersection of the given parabola and the line are $(-1,\frac{1}{4})$ and $(2,1).$

Required area $=\underset{-1}{\overset{2}{\int }}(\frac{x+2}{4}-\frac{x^2}{4})dx$
$={(\frac{(x+2{)}^2}{8}-\frac{x^3}{12})}_{-1}^2$
$=(2-\frac{2}{3})-(\frac{1}{8}+\frac{1}{12})$
$\Rightarrow$ Area $=\frac{9}{8}$ sq. units