Question

Medal for help! Help with finding the area between the curves: x=1-y^2 and x=y^2-1

Answer 1

\[x=1-y^2,y^2=1-x,x=y^2-1,y^2=x+1\]
1-x=x+1
x+x=1-1
2x=0,x=0
\[y^2=1,y=\pm 1,so~ points~ of~ intersection ~are~(0,1)~and~(0,-1)\]
also they cut x-axis at (-1,0) and (1,0)
changing y to -y ,equations remain same.
Hence they are symmetric about x-axis.
\[required~area=2[ \int\limits_{-1}^{0}y~dx+\int\limits_{0}^{1}y~dx ]\]
\[=2\left[ \int\limits_{-1}^{0} \sqrt{x+1}~dx+\int\limits_{0}^{1}\sqrt{1-x}~dx\right]\]
solve it.