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What is a definite integral?

A definite integral computes the signed area under the curve f(x) from x=ax=a to x=bx=b. By the Fundamental Theorem of Calculus, it links antiderivatives to exact area calculations.

Definite integral formula

∫abf(x) dx\int_{a}^{b} f(x)\,dx

According to the Fundamental Theorem: ∫abf(x) dx=F(b)−F(a)\int_{a}^{b} f(x)\,dx = F(b) - F(a), where F′(x)=f(x)F'(x)=f(x).

How to solve a definite integral step by step

Step 1: Identify f(x) and the limits aa and bb.

Step 2: Find an antiderivative F(x)F(x) such that F′(x)=f(x)F'(x)=f(x).

Step 3: Evaluate F(b)−F(a)F(b) - F(a).

Step 4: Simplify to get the exact value.

Example: ∫01x2 dx\int_{0}^{1} x^2\,dx

∫01x2 dx=[x33]01=133−0=13 \int_{0}^{1} x^2\,dx = \left[\tfrac{x^3}{3}\right]_{0}^{1} = \tfrac{1^3}{3} - 0 = \tfrac{1}{3}

Tips & Tricks

  • Ensure f(x) is continuous on [a,b][a,b] for the theorem to apply.
  • Use substitution or integration by parts when the integrand is complex.
  • Split the integral at points where f(x) changes behavior or is discontinuous.

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