Animated Solution for Mathematics - Definite Integration: Let $S(\alpha )=\{(x,y):y^2\le x,0\le x\le \alpha \}$ and $A(\alpha )$ is area of the region $S(\alpha )$. If for $\alpha ,0<\alpha <4$, $A(\alpha ):A(4)=2:5$, then $\alpha$ equals

Imagine you are standing before a coordinate plane, looking at the curve $y^2=x$. This is a rightward-opening parabola, perfectly symmetric about the $x$-axis.

Our task is to find the area of the region $S(\alpha )$, defined by $y^2\le x$ and $0\le x\le \alpha$. This region is the space trapped between the $y$-axis and the vertical line $x=\alpha$, bounded above and below by the parabolic curve.

When we face an integral, the first thing we should look for is symmetry. Because our parabola $y^2=x$ is symmetric about the $x$-axis, the area in the upper half (where $y=\sqrt{x}$) is exactly equal to the area in the lower half (where $y=-\sqrt{x}$).

Instead of dealing with negative square roots, we can simply calculate the area in the first quadrant and multiply it by two. This gives us the elegant setup:

By focusing on the first quadrant, we transform a potentially messy integral into a straightforward application of the power rule.

Now, let us perform the integration. We know that $\sqrt{x}$ is just $x^{1/2}$. Applying the power rule, $\int x^{n}dx=\frac{x^{n+1}}{n+1}$, we get:

Simplifying this, we obtain:

This expression, $\frac{4}{3}{\alpha }^{3/2}$, is the master key to our problem. It tells us exactly how the area grows as we move the boundary $\alpha$ further to the right.

The problem gives us a specific condition: the ratio of $A(\alpha )$ to $A(4)$ is $2:5$. We first find $A(4)$ by substituting $\alpha =4$ into our formula:

Now, we set up the ratio:

The $\frac{1}{3}$ terms cancel out immediately, leaving us with:

Multiplying by $8$, we find ${\alpha }^{3/2}=\frac{16}{5}$.

We are almost at the finish line. We have ${\alpha }^{3/2}=\frac{16}{5}$, and we need to solve for $\alpha$. We raise both sides to the power of $2/3$:

To simplify, we manipulate the exponents:

Since $4^{1/3}=(2^2{)}^{1/3}=2^{2/3}$, we can write this as:

We have arrived at the final result. The beauty of this problem lies in the way symmetry and algebraic manipulation turn a daunting integral into a simple, elegant solution.