6.1 Areas Between Curves/21

${\begin{aligned}x&=1-y^{2},x=y^{2}-1\\[1ex]A&=\int _{a}^{b}[x_{R}-x_{L}]dy=\int _{1}^{-1}[(1-y^{2})-(y^{2}-1)]dy\\[2ex]&=\int _{1}^{-1}[2-2y^{2}]dy=[2y-2({\frac {y^{3}}{3}})|_{-1}^{0}\\[2ex]&=2(1)-2({\frac {(1)^{3}}{3}})-[2(-1)-2({\frac {(-1)^{3}}{3}})]\\[2ex]&=2-{\frac {2}{3}}+2-{\frac {2}{3}}\\[2ex]&={\frac {8}{3}}\end{aligned}}$