JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Area Under Curves

On $[-2,4]$ the graph is $y=\max \{|x|,x|x-2|\}$, and since $|x|\ge 0$ the curve never falls below the x-axis. For $-2\le x\le 0$, $|x|=-x$ exceeds the negative $x(2-x)$, so $y=-x$. For $0\le x\le 1$, $x(2-x)-x=x(1-x)\ge 0$, so $y=x(2-x)$. For $1\le x\le 3$ the line $y=x$ dominates, as both $x(2-x)$ on $[1,2]$ and $x(x-2)$ on $[2,3]$ stay below $x$. For $3\le x\le 4$, $x(x-2)-x=x(x-3)\ge 0$, so $y=x(x-2)$. The area is The area of the region is $12$.