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MathidiaMathematics Academy
Integration
Lesson 37 of 75
Easy practice

Definite integrals

Objective. Evaluate bounded integrals accurately.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

A definite integral evaluates the accumulated change of a function between two limits, calculated as ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f, and because the constants cancel there is no +c in the final numeric answer. First find the antiderivative, then substitute the upper limit b and subtract the result of substituting the lower limit a. For example, ∫₁³ 2x dx = [x²]₁³ = 9 − 1 = 8, representing the net signed area between the curve and the x-axis over that interval. Care is needed with negative areas below the axis, which subtract from the total.

Worked example

Evaluate the definite integral ∫₁⁴ (3x² − 2) dx.

3 … 2
Illustrated card showing 3 … 2
  1. 1Find the antiderivative: F(x) = x³ − 2x.
  2. 2Evaluate at the upper limit: F(4) = 64 − 8 = 56.
  3. 3Evaluate at the lower limit: F(1) = 1 − 2 = −1.
  4. 4Subtract: 56 − (−1) = 57.

Answer: 57

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Common mistake. Students sometimes include +c in a definite integral's final numeric answer, writing 9 − 1 + c instead of simply 8. The constant always cancels exactly when subtracting F(a) from F(b), so it should never appear in the final value.

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