Area Between Curves Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Find the total area enclosed between y=x3−xy = x^3 - x and the xx-axis.

Solution

  1. 1
    Zeros of x3−x=x(x−1)(x+1)x^3-x = x(x-1)(x+1): x=−1,0,1x = -1, 0, 1.
  2. 2
    Sign check: on (−1,0)(-1,0) the function is positive; on (0,1)(0,1) it is negative.
  3. 3
    ∫−10(x3−x) dx=[x44−x22]−10=0−(14−12)=14\int_{-1}^{0}(x^3-x)\,dx = \left[\frac{x^4}{4}-\frac{x^2}{2}\right]_{-1}^{0} = 0-(\frac{1}{4}-\frac{1}{2}) = \frac{1}{4}.
  4. 4
    ∫01(x3−x) dx=14−12=−14\int_{0}^{1}(x^3-x)\,dx = \frac{1}{4}-\frac{1}{2} = -\frac{1}{4} (negative; take absolute value).
  5. 5
    Total area =14+14=12= \frac{1}{4}+\frac{1}{4} = \frac{1}{2}.

Answer

12\frac{1}{2}
When the curve crosses the axis, split at each zero and add absolute values of each piece to get the total geometric area.

About Area Between Curves

The area of the region enclosed between two functions f(x)f(x) and g(x)g(x) from x=ax = a to x=bx = b, computed as A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx.

Learn more about Area Between Curves →

More Area Between Curves Examples

Example 1 easy

Find the area between [formula] and [formula] from [formula] to [formula].

Example 3 easy

Find the area between [formula] and the [formula]-axis where the parabola is above the axis.

Example 4 medium

Find the area between [formula] and [formula].

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