Question

If the area of the closed figure bounded by the following curves $y=$ $x^2$, $y=2-x$, $y=0$ is k. Find 6k

Answer 1

$x^2=2-x\Rightarrow x^2+x-2=0\Rightarrow x^2+2x-x-2=0$

$\Rightarrow (x+2)(x-1)=0$

Area of should region $={\int }_0^1{y}_{1}dx+{\int }_1^2{y}_{2}dx$

$={\int }_0^1{x}^{2}dx+{\int }_1^2(2-x)d$

$=[\frac{x}{3}{]}_0^{\frac{1}{3}}+[2x-\frac{x^2}{2}{]}_1$

$=\frac{1}{3}+2-\left(\frac{3}{2}\right)$

$=\frac{1}{3}+\frac{1}{2}=\frac{5}{6}=k$

$2k=\frac{5}{3}$