Animated Solution for Mathematics - Definite Integration: The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to :-

Imagine you are standing on a coordinate plane, looking at two curves that seem to be reaching out to embrace each other. We have $y^2=8x$, a classic parabola opening its arms to the right, starting from the origin.

Then, we have $y^2=16(3-x)$, a second parabola, but this one is different; it opens to the left, with its vertex anchored at $x=3$. These two curves trap a beautiful, symmetric region between them.

Before we can measure the area, we must know where these two curves meet. Since both equations are defined by $y^2$, we can set them equal to each other:

Dividing both sides by $8$, we get $x=2(3-x)$, which expands to $x=6-2x$. Solving for $x$, we find $3x=6$, or simply $x=2$.

Now, we need the $y$-coordinates. Substituting $x=2$ into $y^2=8x$, we get $y^2=16$, which means $y=\pm 4$. So, our dancers meet at the points $(2,4)$ and $(2,-4)$.

Now, we face a choice: how to integrate? We could use vertical strips, but that would force us to split the integral into two parts because the upper and lower boundaries change at $x=2$.

Instead, let's use horizontal strips. By integrating with respect to $y$, we can treat the entire region as one continuous shape.

Because the region is perfectly symmetric about the $x$-axis, we can calculate the area of the top half (from $y=0$ to $y=4$) and simply double it. The total area is given by:

We need to express $x$ in terms of $y$ for both curves. From $y^2=8x$, we get $x_{left}=\frac{y^2}{8}$. From $y^2=16(3-x)$, we rearrange to find $x_{right}=3-\frac{y^2}{16}$.

Now, let's set up our integral:

Simplifying the integrand, we combine the $y^2$ terms: $-\frac{y^2}{16}-\frac{2y^2}{16}=-\frac{3y^2}{16}$. Our integral becomes:

Integrating term by term, the integral of $3$ is $3y$, and the integral of $\frac{3y^2}{16}$ is $\frac{y^3}{16}$. Evaluating from $0$ to $4$, we get:

Plugging in the upper limit:

The area is exactly $16$ square units. It is a clean, elegant result that rewards our careful geometric thinking.