Question

Let for $x\in R$.
$f(x)=\frac{x+|x|}{2}\text{ and }g(x)=\{\begin{array}{ll}x,& x<0\\ x^2,& x\ge 0\end{array}$
Then area bounded by the curve $y=(f\circ g)(x)$ and the lines $y=0,2y-x=15$ is equal to

Answer 1

Answer: 72

$2y-x=15$
$A=\underset{0}{\overset{3}{\int }}\left(\frac{x+15}{2}-x^2\right)dx+\frac{1}{2}\times \frac{15}{2}\times 15$
$\frac{x^2}{4}+\frac{15x}{2}-{\frac{x^3}{3}|}_0^3+\frac{225}{4}$
$=\frac{9}{4}+\frac{45}{2}-9+\frac{225}{4}=\frac{99-36+225}{4}$
$=\frac{288}{4}=72$