Area between curves
Area Between Curves
Area Between Curves is a topic in Applications of Integration in the California Calculus Standards. It is aligned to Standard 16.0, which requires students to use definite integrals in problems involving area, including the area enclosed between two curves.
The area between two curves is \(\int_a^b(\text{top}-\text{bottom})\,dx\), integrated between the points where the curves intersect.
Theory
The area between two curves is the integral of the top curve minus the bottom curve, \(\displaystyle\int_a^b (f-g)\,dx\). The limits \(a\) and \(b\) come from the intersection points of the two curves.
When two curves enclose a region, its area is built from thin vertical strips. Each strip has height top curve minus bottom curve and width \(dx\):
where \(f\) is on top and \(g\) on the bottom throughout \([a,b]\).
The limits \(a\) and \(b\) are the intersection points, found by setting \(f(x)=g(x)\).
The area between curves, and the limits:
How to find the area between two curves
- Find the intersections by solving \(f(x)=g(x)\) for the limits.
- Decide which curve is on top by testing a point between them.
- Integrate \(\int_a^b(\text{top}-\text{bottom})\,dx\).
They meet at \(x=0,1\); on \((0,1)\), \(x>x^2\), so integrate top \(-\) bottom.
| \(\int_0^1 (x-x^2)\,dx\) | \(=\) | \(\left[\dfrac{x^2}{2}-\dfrac{x^3}{3}\right]_0^1\) |
| \(=\) | \(\dfrac{1}{2}-\dfrac{1}{3}\) | |
| \(=\) | \(\dfrac{1}{6}\) |
They meet where \(x^2-x-2=0\), i.e. \(x=-1,2\); the line is on top.
| \(\int_{-1}^{2}(x+2-x^2)\,dx\) | \(=\) | \(\left[\dfrac{x^2}{2}+2x-\dfrac{x^3}{3}\right]_{-1}^{2}\) |
| \(=\) | \(\dfrac{10}{3}-\left(-\dfrac{7}{6}\right)\) | |
| \(=\) | \(\dfrac{9}{2}\) |
They meet at \(x=\pm 2\); \(4\) is on top.
| \(\int_{-2}^{2}(4-x^2)\,dx\) | \(=\) | \(\left[4x-\dfrac{x^3}{3}\right]_{-2}^{2}\) |
| \(=\) | \(\dfrac{16}{3}-\left(-\dfrac{16}{3}\right)\) | |
| \(=\) | \(\dfrac{32}{3}\) |
Integrate the vertical gap, top minus bottom.
| \(\text{Area}\) | \(=\) | \(\int_a^b (f-g)\,dx\) |
Common pitfalls
Frequently asked questions
How do you find the area between two curves?
Integrate the top curve minus the bottom curve between their intersection points: \(\int_a^b (f-g)\,dx\).
How do you find the limits of integration?
Set the two curves equal, \(f(x)=g(x)\), and solve. The solutions are the \(x\)-values where the region begins and ends.
How do you know which curve is on top?
Test a point between the intersections. Whichever function is larger there is the top curve on that interval.
What if the curves cross inside the interval?
Split the integral at the crossing point and use the correct top-minus-bottom order on each piece.