Question

The area common to the parabola $y=2x^2$ and $y=x^2+4$

• A. $\frac{2}{3}sq.units$
• B. $\frac{3}{2}sq.units$
• C. $\frac{32}{3}sq.units$
• D. none of these.

Answer 1

The correct option is C $\frac{32}{3}sq.units$

Find the P.O.I of the parabolas equate the equations $y=2x^2$ and $y=x^2+y$ we get

$2x^2=x^2+4$

$x^2=4$

$x=\pm 2$

$y=8$

$\therefore$ POI are $A(-2,8)$ & $C(2,8)$

$\therefore$ the required are ABCD

$A={\int }_{-2}^2(y_1-y_2)dx$ (where $y_1=x^2+4\&y_2=2x^2$)

$={\int }_{-2}^2(x^2+4-2x^2)dx={\int }_{-2}^2(4-x^2)dx={(4x-\frac{x^3}{3})}_{-2}^2$

$\left[4\right(2)-\frac{(2{)}^3}{3}]-\left[4\right(-2)-\frac{(-2{)}^3}{3}]=[8-\frac{8}{3}]-[8+\frac{8}{3}]$

$=\frac{32}{3}$ sq. units