To find the area between two curves using integrals, we can follow these simple steps:

1. Identify the Curves: First, we look at two functions, which we’ll call $f(x)$ and $g(x)$. We need to make sure that $f(x)$ is always above $g(x)$ in a certain range from $[a, b]$.

2. Find the Points Where They Meet: We then figure out where these curves cross each other. This is done by solving the equation $f(x) = g(x)$. The points where they meet, called $x_1$ and $x_2$, will help us set the limits for our calculation.

3. Set Up the Integral: The area $A$ between the curves from point $x_1$ to point $x_2$ is calculated with this formula:

   $$A = \int_{x_1}^{x_2} (f(x) - g(x)) \, dx$$

   Here, we are finding the space between the top curve and the bottom curve by adding up tiny slices from $x_1$ to $x_2$.

4. Evaluate the Integral: After setting it up, we often use numerical methods or special tools to help calculate the integral, especially if the functions are a bit tricky.

Example:

Let’s use the functions $f(x) = x^2$ and $g(x) = x$. To find out where these curves meet, we solve $x^2 = x$, which gives us the points $x = 0$ and $x = 1$.

Now, we can find the area between them:

$$A = \int_{0}^{1} (x^2 - x) \, dx$$

This part breaks down as follows:

$$A = \left[ \frac{x^3}{3} - \frac{x^2}{2} \right]_{0}^{1}$$

When we plug in the numbers, we get:

$$A = \frac{1}{3} - \frac{1}{2} = -\frac{1}{6}$$

This negative value shows we need to make sure we're using the right top and bottom functions to avoid mistakes with our area calculation.