Animated Solution for Mathematics - Definite Integration: The area of the region $\{(x,y):0\le x\le \frac{9}{4},0\le y\le 1,x\le 3y,x+y\ge 2\}$ is

Welcome, future engineer. Today we are not just solving a problem; we are mapping a territory. Imagine you are standing on a coordinate plane with boundaries defined by the vertical wall at $x=\frac{9}{4}$, the ground at $y=0$, the steep slope of $x=3y$, and the diagonal barrier of $x+y=2$.

Our mission is to calculate the area of the quadrilateral $PQRS$ trapped within these lines.

Mapping the Vertices

First, we must find the vertices of the quadrilateral. Point $P$ is the intersection of $x+y=2$ and $x=3y$. Solving this system gives $y=\frac{1}{2}$ and $x=\frac{3}{2}$, so $P=(\frac{3}{2},\frac{1}{2})$.

Next, we find $Q$ and $R$ on the x-axis. $Q$ is where $x+y=2$ hits the axis, yielding $Q=(2,0)$. $R$ is where $x=\frac{9}{4}$ hits the axis, yielding $R=(\frac{9}{4},0)$.

Finally, $S$ is where $x=\frac{9}{4}$ meets $x=3y$. Substituting $x=\frac{9}{4}$ into $x=3y$ gives $y=\frac{3}{4}$, so $S=(\frac{9}{4},\frac{3}{4})$.

The Secret Weapon

Now, we deploy the Shoelace Formula. It is elegant, powerful, and foolproof. We arrange our vertices in order: $P(\frac{3}{2},\frac{1}{2})$, $Q(2,0)$, $R(\frac{9}{4},0)$, and $S(\frac{9}{4},\frac{3}{4})$.

We calculate the determinants of the consecutive coordinate pairs:

Evaluating these determinants individually:

1. The first determinant: $(\frac{3}{2}\times 0)-(2\times \frac{1}{2})=-1$. 2. The second determinant: $(2\times 0)-(\frac{9}{4}\times 0)=0$. 3. The third determinant: $(\frac{9}{4}\times \frac{3}{4})-(0\times \frac{9}{4})=\frac{27}{16}$. 4. The fourth determinant: $(\frac{9}{4}\times \frac{1}{2})-(\frac{3}{4}\times \frac{3}{2})=\frac{9}{8}-\frac{9}{8}=0$.

Summing these values gives:

Multiplying by the factor of $\frac{1}{2}$ from the Shoelace Formula, we arrive at the final area:

You have conquered the geometry. Remember, in JEE Advanced, the complexity is often just a mask for a simple, beautiful structure waiting to be revealed.