Animated Solution for Mathematics - Definite Integration: The area of the region $S=\{(x,y):3x^2\le 4y\le 6x+24\}$ is .

To determine the area of the region $S$ trapped between the parabola $3x^2\le 4y$ and the line $4y\le 6x+24$, we first express the boundaries as functions of $x$.

The lower boundary is defined by the parabola:

The upper boundary is defined by the line:

Where the Walls Meet

To find the limits of integration, we identify the points where the two curves intersect by setting the expressions for $y$ equal to each other:

Dividing the entire equation by $3$, we obtain:

Rearranging this into the standard quadratic form yields:

Factoring the quadratic equation gives $(x-4)(x+2)=0$. Thus, the garden spans the interval from $x=-2$ to $x=4$.

Setting up the Integral

The area $A$ between two curves is calculated by integrating the difference between the upper function and the lower function over the determined interval.

We set up the integral as follows:

To simplify the calculation, we factor out the constant $\frac{1}{4}$:

We now perform the integration term by term:

Next, we evaluate the expression at the upper limit ($x=4$) and the lower limit ($x=-2$):

For $x=4$:

For $x=-2$:

Finally, we subtract the value at the lower limit from the value at the upper limit:

The final area of the region $S$ is 27 square units.