Area Between Curves Math Example 1

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

Example 1

easy
Find the area between f(x)=x+2f(x) = x+2 and g(x)=x2g(x) = x^2 from x=−1x=-1 to x=2x=2.

Solution

  1. 1
    Intersections: x+2=x2⇒(x−2)(x+1)=0⇒x=−1,2x+2=x^2 \Rightarrow (x-2)(x+1)=0 \Rightarrow x=-1,2 (endpoints).
  2. 2
    At x=0x=0: f(0)=2>g(0)=0f(0)=2 > g(0)=0, so f≥gf \geq g throughout.
  3. 3
    A=∫−12(x+2−x2) dx=[x22+2x−x33]−12A = \int_{-1}^{2}(x+2-x^2)\,dx = \left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^{2}.
  4. 4
    F(2)=2+4−83=103F(2) = 2+4-\frac{8}{3} = \frac{10}{3}; F(−1)=12−2+13=−76F(-1)=\frac{1}{2}-2+\frac{1}{3}=-\frac{7}{6}.
  5. 5
    A=103+76=206+76=276=92A = \frac{10}{3}+\frac{7}{6} = \frac{20}{6}+\frac{7}{6} = \frac{27}{6} = \frac{9}{2}.

Answer

92\frac{9}{2}
Identify intersection points, confirm which function is on top, then integrate the difference over the interval.

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 2 hard

Find the total area enclosed between [formula] and the [formula]-axis.

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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