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Calculus, Matrices & Vectors · SS 3

Integration: power rule

Reverse differentiation, and find the area under a curve.

  • ∫ xn dx = xn + 1n + 1 + C (n ≠ −1)
  • ∫ab f(x) dx = F(b) − F(a)

What the letters mean

C
the constant of integration
a, b
the limits (start and end x-values)
F
the integrated function

Worked example

Evaluate ∫03 x2 dx.

  1. Step 1: Integrate

    ∫ x2 dx = x33

  2. Step 2: Top limit minus bottom limit

    333 − 033 = 9 − 0 = 9

Final answer: 9

Add 1 to the power, then divide by the new power.

See it

Play with this picture from the Integration lesson.

ab

4 strips add up to ≈ 8.859

∫ (x2) dx = 13x3 + c, so F(x) = 13x3

∫03 (x2) dx = F(3) − F(0) = 9 − 0 = 9

Lower limit a0
Upper limit b3
Number of strips4