Evaluate the following integral ∫ from u to v of f(x) dx

To evaluate the integral ∫_u^v f(x) dx, we start by identifying the function f(x) that we need to integrate. The process involves several steps:

1. Find the antiderivative: First, we need to determine an antiderivative F(x) of f(x). This means we must find a function F(x) such that F'(x) = f(x).
2. Apply the Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x), then:

   ∫_u^v f(x) dx = F(v) – F(u).

3. Substitute the limits: After finding F(v) and F(u), we simply substitute these values into the equation above. This will give us the value of the integral over the interval [u, v].

Let’s take an example:

If f(x) = x², then the antiderivative F(x) = (1/3)x³.

So, we would compute:

∫uv x² dx = F(v) - F(u) = (1/3)v³ - (1/3)u³.

By following these steps, you can evaluate any definite integral as long as you know the function you are working with, along with your limits of integration.