Animated Solution for Mathematics - Definite Integration: The area (in sq. units) of the region $\{(x,y)\in R^2:x^2\le y\le 3-2x\}$, is:

Imagine you are standing on a coordinate plane. To your left, a graceful parabola, $y=x^2$, curves upward like a valley. To your right, a straight line, $y=3-2x$, cuts across the landscape like a mountain ridge.

The region we are interested in is the 'pocket' of space trapped between these two. It is a finite, beautiful, and perfectly defined slice of the plane. Our goal today is to measure its area.

Before we can measure the area, we must know where the 'fencing' begins and ends. These are the intersection points where the $y$-values are identical. We set them equal:

By moving everything to one side, we arrive at the quadratic equation:

Factoring this quadratic requires two numbers that multiply to $-3$ and add to $2$. Those numbers are $3$ and $-1$. Thus, we factor it as:

Our boundaries are locked: the region begins at $x=-3$ and ends at $x=1$.

Now, we build our integral. The area $A$ between two curves is the accumulation of the vertical distance between them. We define this as the integral of the 'upper' function minus the 'lower' function:

In our case, the line $y=3-2x$ sits above the parabola $y=x^2$ within our interval. So, our integral becomes:

This is the heart of the problem. We are essentially summing up an infinite number of tiny vertical strips, each with height $(3-2x-x^2)$ and width $dx$.

Now, we perform the integration. Using the power rule, $\int x^{n}dx=\frac{x^{n+1}}{n+1}$, we integrate term by term:

We evaluate this from $-3$ to $1$. First, we plug in the upper limit, $x=1$:

Next, we plug in the lower limit, $x=-3$:

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

Converting $9$ to a fraction with a denominator of $3$, we get $\frac{27}{3}$. Adding these together, we find the total area is exactly:

Look at what we have achieved. We took two abstract equations, found their intersection, set up a definite integral, and calculated the precise space they enclose.

This is the power of calculus—it allows us to quantify the world around us with absolute precision. Keep this logic in your toolkit; whether it is a parabola and a line or two complex trigonometric functions, the process remains the same. You have mastered the dance of the curves!