Question

Find the area of the region bounded by the graph of y^2 = -4(x-1) and y^2 = -2(x-2). Provide a graph.

Answer 1

\[y^2=-4(x-1)\quad,\quad x=1-\frac{y^2}{4}\]\[y^2=-2(x-2)\quad,\quad x=2-\frac{y^2}{2}\]\[-4(x-1)=-2(x-2)\quad,\quad x=0\quad\Rightarrow\quad y=\pm 2\]\[S=\int\limits_{-2}^{2}\left(2-\frac{y^2}{2}-1+\frac{y^2}{4}\right)dy=\int\limits_{-2}^{2}\left(1-\frac{y^2}{4}\right)dy=\int\limits_{0}^{2}\left(2-\frac{y^2}{2}\right)dy=\]\[=\left(2y-\frac{y^3}{6}\right)|_0^2=4-\frac{4}{3}=\frac{8}{3}\] http://www.wolframalpha.com/input/?i=y%5E2%3D-4%28x-1%29+and+y%5E2%3D-2%28x-2%29