How to set up a definite integral to find the area under a curve?

To set up a definite integral to find the area under a curve, you'll need to follow these steps:

Identify the function and the interval: Let's call the function whose area we want to find 'f(x)' and the interval [a, b]. The function should be continuous and integrable over this interval.
Understand the limits of integration: The numbers 'a' and 'b' are called the limits of integration. The definite integral will give you the signed area between the curve and the x-axis over the interval [a, b]. If 'a' > 'b', the integral will give the signed area between the curve and the x-axis over the interval [b, a].
Write the definite integral: Using the function and the limits of integration, write the definite integral as follows:
```
∫ from a to b f(x) dx
```
Here, '∫' is the integral sign, 'from a to b' are the limits of integration, 'f(x)' is the function, and 'dx' indicates that 'x' is the variable of integration.
Evaluate the definite integral: To find the signed area, you'll need to evaluate the definite integral. This involves finding the antiderivative (indefinite integral) of 'f(x)', denoted as 'F(x)', and then applying the Fundamental Theorem of Calculus:
```
∫ from a to b f(x) dx = F(b) - F(a)
```
This will give you the signed area between the curve and the x-axis over the interval [a, b]. If the result is positive, the area is above the x-axis; if it's negative, the area is below the x-axis.