Definite Integral Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Definite Integral.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An integral evaluated between specific lower and upper bounds, yielding a single numerical value rather than a function.

The signed total area under the curve from a to b—positive above the x-axis, negative below.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Definite integral gives a number; indefinite integral gives a function.

Common stuck point: Area below the x-axis counts as negative — if you need total geometric area, integrate the absolute value.

Sense of Study hint: Write out F(b) - F(a) step by step, substituting each bound separately before subtracting.

Worked Examples

Example 1

easy
Evaluate \int_1^4 (2x - 1)\,dx.

Solution

  1. 1
    Find the antiderivative: F(x) = x^2 - x.
  2. 2
    Apply the Fundamental Theorem: \int_1^4 (2x-1)\,dx = F(4) - F(1).
  3. 3
    Compute F(4) = 16 - 4 = 12 and F(1) = 1 - 1 = 0.
  4. 4
    Result: 12 - 0 = 12.

Answer

12
For a definite integral, find the antiderivative, evaluate it at the upper bound, then subtract its value at the lower bound. The constant of integration cancels out when you subtract, so it can be omitted.

Example 2

medium
Evaluate \int_{-1}^{2} (x^2 - x)\,dx and interpret the sign of the result.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate \int_0^3 x^2\,dx.

Example 2

hard
Evaluate \int_0^{\pi} \sin x\,dx and explain the geometric meaning.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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