Animated Solution for Mathematics - Definite Integration: The area of the region $\{(x,y):y^2\le 4x,x<4,\frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,x\ne 3\}$ is

Welcome, fellow traveler of the JEE landscape. Today, we are not just solving an integral; we are mapping a territory.

Imagine you are standing on a coordinate plane, looking at a right-opening parabola defined by $y^2=4x$. This is our canvas.

We are constrained by a vertical boundary at $x=4$ and a rational inequality: This inequality is the gatekeeper of our region, dictating where we can and cannot walk.

Let us simplify the chaos. If we define $g(x)=\frac{x(x-1)(x-2)}{(x-3)(x-4)}$, our condition becomes $y\cdot g(x)>0$.

This is a logical switch. For the product to be positive, $y$ and $g(x)$ must share the same sign.

If $g(x)$ is positive, $y$ must be positive (the upper half of the parabola). If $g(x)$ is negative, $y$ must be negative (the lower half).

To find where $g(x)$ is positive or negative, we use the Wavy Curve Method. Our critical points are $x=0,1,2,3,4$.

By testing intervals, we find the sign of $g(x)$ alternates: Positive in $(0,1)$ Negative in $(1,2)$ Positive in $(2,3)$ Negative in $(3,4)$

Our shaded region is a series of alternating strips. In $(0,1)$, we shade the upper half; in $(1,2)$, we shade the lower half; in $(2,3)$, we return to the upper half; and in $(3,4)$, we finish with the lower half.

Now, here is the moment of brilliance. A novice would calculate four separate integrals, but you see the symmetry.

The parabola $y^2=4x$ is perfectly symmetric about the $x$-axis. The area of the upper half is identical to the area of the lower half.

Whether we are shading the upper strip or the lower strip, the area contribution is the same. Therefore, the total area $A$ is simply the integral of the upper curve $y=2\sqrt{x}$ from $x=0$ to $x=4$.

We are left with the integral:

We rewrite $\sqrt{x}$ as $x^{1/2}$ and apply the power rule of integration:

Our integral becomes:

Substituting the limits, we get:

Since $4^{3/2}=8$, the final area is:

We have navigated the constraints, mastered the signs, utilized symmetry, and conquered the calculus. The result, $\frac{32}{3}$, is the proof of your analytical prowess.