JEE Main 2023 (Online) 31st January Morning ShiftQuestion 1
Let for $x\in\mathbb R$, $f(x)=\frac{x+|x|}{2}$ and $g(x)=\begin{cases}x, & x<0\\x^{2}, & x\ge0\end{cases}$. Then the area bounded by the curve $y=(f\circ g)(x)$ and the lines $y=0$, $2y-x=15$ is equal to ______.
Application of Integrals
Answer$72$
Solution
$f\circ g=0$ for $x<0$ and $x^2$ for $x\ge0$. Line $y=\frac{x+15}2$ meets $y=x^2$ at $x=3$. Area $=\int_{-15}^0\frac{x+15}2dx+\int_0^3\left(\frac{x+15}2-x^2\right)dx=\frac{225}4+\frac{63}4=72$.