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Definite integrals and areas

A definite integral has limits and gives a number. It measures the area between a curve and the x-axis.

Evaluating

∫a^{b} f(x) dx = [F(x)]a^{b} = F(b) - F(a), where F is the integral of f. There is no + c: it would cancel.

Area under a curve

For a curve above the x-axis between x = a and x = b, the area is ∫a^{b} y dx.

Areas below the axis

Where the curve is below the x-axis, the integral is negative. To find a total area, find where the curve crosses the axis, integrate each part separately and add the sizes (ignore the signs).

Worked example

Evaluate ∫1^{3} (3x2 - 2x) dx.

  1. Integrate: [x3 - x2]
  2. Upper limit: 27 - 9 = 18
  3. Lower limit: 1 - 1 = 0

Answer: 18 - 0 = 18

Key idea: ∫a^{b} f(x) dx = F(b) - F(a). Areas below the x-axis give negative integrals: split at the roots and add the sizes.

Check yourself

  1. Evaluate ∫0^{2} (x2 + 1) dx. Give your answer as a fraction in its simplest form.
    Show answer

    14/3. The integral is x3/3 + x. From 0 to 2: (8/3 + 2) - 0 = 14/3.

  2. Evaluate ∫1^{4} 6sqrt(x) dx.
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    28. 6x1/2 integrates to 4x3/2. 4(8) - 4(1) = 28.

  3. Evaluate ∫1^{2} 2/x2 dx.
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    1. 2x-2 integrates to -2x-1. (-1) - (-2) = 1.

  4. Find the area enclosed by y = x(x - 4) and the x-axis. Give your answer as a fraction in its simplest form.
    Show answer

    32/3. ∫0^{4} (x2 - 4x) dx = 64/3 - 32 = -32/3. The region is below the axis, so the area is 32/3.

  5. ∫0^{4} (x2 - 4x) dx = -32/3. What does the negative sign tell you?
    • You made an error, as areas cannot be negative
    • The curve has no roots
    • The region is below the x-axis
    • The limits are the wrong way round
    Show answer

    The region is below the x-axis. Integrals of regions below the x-axis are negative. The area is the size, 32/3.

  6. Find the total area between y = x2 - 1, the x-axis and the lines x = 0 and x = 2.
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    2. From 0 to 1: ∫ = 1/3 - 1 = -2/3 (area 2/3). From 1 to 2: ∫ = (8/3 - 2) - (1/3 - 1) = 4/3. Total area = 2/3 + 4/3 = 2.

  7. Evaluate ∫-1^{1} (x3 + x2) dx. Give your answer as a fraction in its simplest form.
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    2/3. The integral is x4/4 + x3/3. From -1 to 1: (1/4 + 1/3) - (1/4 - 1/3) = 2/3.

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Last updated 7 November 2026. Spotted a mistake? Tell us.