Answer to Question #317119 in Civil and Environmental Engineering for Rodz

Question

Find the area between the curves y^2 - y + x = 0 and y^2 = x + 3.

Answer 1

$A=\int _a^b|f\left(x\right)-g\left(x\right)|dx$

$=\int _{-3}^{-2}\left|f_3\left(x\right)-f_4\left(x\right)\right|dx+\int _{-2}^{-\frac{3}{4}}\left|f_2\left(x\right)-f_3\left(x\right)\right|dx+\int _{-\frac{3}{4}}^{\frac{1}{4}}\left|f_1\left(x\right)-f_2\left(x\right)\right|dx$

$=\int _{-3}^{-2}\left|\sqrt{x+3}-\left(-\sqrt{x+3}\right)\right|dx+\int _{-2}^{-\frac{3}{4}}\left|\frac{1-\sqrt{1-4x}}{2}-\sqrt{x+3}\right|dx+\int _{-\frac{3}{4}}^{\frac{1}{4}}\left|\frac{1+\sqrt{1-4x}}{2}-\frac{1-\sqrt{1-4x}}{2}\right|$

$=\frac{4}{3}+\frac{61}{24}+\frac{4}{3}$

$=\frac{125}{24}$

A= 5.2083