Question

Draw a rough sketch of $y^2=x+1$ and $y^2=-x+1$ and determine the area enclosed by the two curves.

Answer 1

Given curves are $y^2=x+1$ .....(1)

And, $y^2=-x+1$ .......(2)

On solving equation 1 and 2, we get,

$x+1=-x+1$

$2x=0⟹x=0$

On putting $x=0$ in equation 1, we get,

$y=\pm 1$

Therefore, two given parabolas intersect each other at A(0,1) and B(0,-1) as shown in the figure.

Therefore,

Required area = 2[ Area of the region OCAO + Area of the region ODAO ]

$=2[\underset{-1}{\overset{0}{\int }}\sqrt{x+1}dx+\underset{0}{\overset{1}{\int }}\sqrt{1-x}dx]$

$=2\times \frac{2}{3}({\left[\right(x+1{)}^{3/2}{]}_{-1}}^0-{\left[\right(1-x{)}^{3/2}{]}_0}^1)$

$=\frac{4}{3}(1+1)=\frac{8}{3}sq.units$