Definite Integral Math Example 2

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Example 2

medium
Evaluate ∫−12(x2−x) dx\int_{-1}^{2} (x^2 - x)\,dx and interpret the sign of the result.

Solution

  1. 1
    Antiderivative: F(x)=x33−x22F(x) = \frac{x^3}{3} - \frac{x^2}{2}.
  2. 2
    Evaluate at upper bound: F(2)=83−42=83−2=23F(2) = \frac{8}{3} - \frac{4}{2} = \frac{8}{3} - 2 = \frac{2}{3}.
  3. 3
    Evaluate at lower bound: F(−1)=−13−12=−26−36=−56F(-1) = \frac{-1}{3} - \frac{1}{2} = -\frac{2}{6} - \frac{3}{6} = -\frac{5}{6}.
  4. 4
    Result: 23−(−56)=46+56=96=32\frac{2}{3} - \left(-\frac{5}{6}\right) = \frac{4}{6} + \frac{5}{6} = \frac{9}{6} = \frac{3}{2}.

Answer

32\frac{3}{2}
The positive result means the net signed area above the x-axis exceeds the area below it on [−1,2][-1, 2]. Note that the function x2−xx^2-x dips below the axis on (0,1)(0,1), contributing negative area, but the overall integral is positive.

About Definite Integral

An integral evaluated between specific bounds aa and bb, yielding a single number: the signed area under the curve.

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More Definite Integral Examples

Example 1 easy

Evaluate [formula].

Example 3 easy

Evaluate [formula].

Example 4 hard

Evaluate [formula] and explain the geometric meaning.

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