Area Between Curves Formula
The Formula
If the curves cross, split into separate integrals at each intersection point.
When to use: 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.
Quick Example
A = \int_0^1 (x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}
Notation
What This Formula Means
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.
Formal View
Worked Examples
Example 1
easySolution
- 1 Intersections: x+2=x^2 \Rightarrow (x-2)(x+1)=0 \Rightarrow x=-1,2 (endpoints).
- 2 At x=0: f(0)=2 > g(0)=0, so f \geq g throughout.
- 3 A = \int_{-1}^{2}(x+2-x^2)\,dx = \left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^{2}.
- 4 F(2) = 2+4-\frac{8}{3} = \frac{10}{3}; F(-1)=\frac{1}{2}-2+\frac{1}{3}=-\frac{7}{6}.
- 5 A = \frac{10}{3}+\frac{7}{6} = \frac{20}{6}+\frac{7}{6} = \frac{27}{6} = \frac{9}{2}.
Answer
Example 2
hardCommon Mistakes
- Not finding intersection points: if you don't know where the curves cross, you may integrate with the wrong function on top, getting a negative (and incorrect) area.
- Forgetting absolute value or not splitting the integral when curves cross: if f is above g on part of the interval and below on another, a single integral without absolute value will give cancellation and the wrong answer.
- Integrating with respect to the wrong variable: sometimes integrating with respect to y (horizontal slices) is much simpler. If the region is easier to describe as 'right minus left,' switch to dy integration.
Why This Formula Matters
This technique is a building block for volumes of revolution, work problems, and probability (area under probability density curves). It's one of the most common applications of integration.
Frequently Asked Questions
What is the Area Between Curves formula?
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.
How do you use the Area Between Curves formula?
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.
What do the symbols mean in the Area Between Curves formula?
Top minus bottom (for vertical slices) or right minus left (for horizontal slices with dy integration).
Why is the Area Between Curves formula important in Math?
This technique is a building block for volumes of revolution, work problems, and probability (area under probability density curves). It's one of the most common applications of integration.
What do students get wrong about Area Between Curves?
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.
What should I learn before the Area Between Curves formula?
Before studying the Area Between Curves formula, you should understand: definite integral, fundamental theorem.
Want the Full Guide?
This formula is covered in depth in our complete guide:
How to Integrate Rational Functions: Long Division and Partial Fractions →