When we want to find the area between two curves, we need to set up the integrals the right way.

This means we have to figure out the two functions we're looking at, usually called $f(x)$ and $g(x)$. One of these functions will be on top of the other over the range we're interested in.

To find the area $A$ between the two curves from $x=a$ to $x=b$, we use this formula:

$$A = \int_{a}^{b} (f(x) - g(x)) \, dx$$

Finding Where the Curves Meet

Before we can use this formula, we need to find where the two curves intersect. These points are found by solving the equation $f(x) = g(x)$. The solutions to this equation give us the points $a$ and $b$, which are the limits for our integral.

Calculating the Area

After we find out which function is on top ($f(x)$) and which is on the bottom ($g(x)$), we can calculate the area. If the curves are oriented vertically, we might also need to look at the functions on the left and right sides in terms of $y$. In this case, the formula looks like this:

$$A = \int_{c}^{d} (R(y) - L(y)) \, dy$$

Here, $R(y)$ is the right function and $L(y)$ is the left function.

Example

For example, if we want to find the area between the curves $y = x^2$ and $y = x + 2$, we start by solving for where they intersect:

$$x^2 = x + 2 \implies x^2 - x - 2 = 0 \implies (x-2)(x+1) = 0 \Rightarrow x = -1, 2$$

Next, we set up our integral:

$$A = \int_{-1}^{2} ((x + 2) - (x^2)) \, dx$$

Calculating this integral will give us the area between the two curves from $x = -1$ to $x = 2$.