How to evaluate a definite integral using the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus (FTC) provides a way to evaluate definite integrals using antiderivatives, making calculations much simpler. Here's how to use it:

  1. Understand the theorem: The FTC states that if a function f(x) is continuous on the interval [a, b], and F(x) is an antiderivative of f(x) on (a, b), then the definite integral of f(x) from a to b is equal to F(b) - F(a).

  2. Find the antiderivative: To evaluate the definite integral using the FTC, first find an antiderivative F(x) of the function f(x) whose integral you're calculating. This is typically the most challenging part of the process.

  3. Apply the FTC: Once you've found F(x), evaluate F(x) at the upper limit b and the lower limit a. Then, subtract the two results: ∫ from a to b f(x) dx = F(b) - F(a).

Here's a simple example: Evaluate ∫ from 1 to 4 (3x^2 - 2x + 1) dx.

  • Find the antiderivative F(x) of f(x) = 3x^2 - 2x + 1. The antiderivative is F(x) = x^3 - x^2 + x.
  • Evaluate F(x) at the upper limit 4 and the lower limit 1: F(4) = 64 - 16 + 4 = 52 and F(1) = 1 - 1 + 1 = 1.
  • Subtract the two results: ∫ from 1 to 4 (3x^2 - 2x + 1) dx = F(4) - F(1) = 52 - 1 = 51.
How to evaluate a definite integral using the Fundamental Theorem of Calculus? — Calculus Basics | Unlo