JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Area Under Curves

Split the region by $x$.

For $x\in [1,2]$, the binding constraints are $y\ge x-1$ (from $y\ge |x-1|$) and $y\le 3-x$ (from $y+|x|\le 3$), since here $3-x\le \frac{x^2+3}{2}$; the strip has width $4-2x$, contributing


For $x\in [0,1]$, the binding upper bound switches to the parabola, $y\le \frac{x^2+3}{2}$, while the lower bound stays $y\ge 1-x$; the width is $\frac{(x+1{)}^2}{2}$, contributing


For $x\in [-1,0]$, the same parabola bound $y\le \frac{x^2+3}{2}$ stays below $3+x$, and the lower bound is $1-x$; the width is again $\frac{(x+1{)}^2}{2}$, contributing


For $x<-1$, $1-x$ exceeds $3+x$, so the region is empty there.

Adding the pieces,