Area Between Curves Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Area Between Curves.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

To find the area between two curves, subtract the lower curve from the upper curve and integrate. It's like finding the area under the top curve and subtracting the area under the bottom curve—the difference is the area of the 'sandwich' between them.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Area between curves extends the definite integral from 'area under a curve' to 'area between curves.' The key is correctly identifying which function is on top and splitting at intersection points.

Common stuck point: When curves cross, the 'top' and 'bottom' functions swap. You must find the intersection points and set up separate integrals for each sub-interval, or use absolute value.

Sense of Study hint: Set the two functions equal to find intersection points first, then test a value in each sub-interval to determine which is on top.

Worked Examples

Example 1

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

Solution

  1. 1
    Intersections: x+2=x^2 \Rightarrow (x-2)(x+1)=0 \Rightarrow x=-1,2 (endpoints).
  2. 2
    At x=0: f(0)=2 > g(0)=0, so f \geq g throughout.
  3. 3
    A = \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-\frac{8}{3} = \frac{10}{3}; F(-1)=\frac{1}{2}-2+\frac{1}{3}=-\frac{7}{6}.
  5. 5
    A = \frac{10}{3}+\frac{7}{6} = \frac{20}{6}+\frac{7}{6} = \frac{27}{6} = \frac{9}{2}.

Answer

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

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the area between y = 4-x^2 and the x-axis where the parabola is above the axis.

Example 2

medium
Find the area between y = x^2 and y = 2x.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

definite integralfundamental theorem