Integrals · Definition
Area between curves
Before you start you need to know
When two curves bound a region, its area is found by integrating the difference between the upper curve and the lower curve.
Definition
The area between the graphs of \(f\) and \(g\) on \([a,b]\), with \(f\ge g\), is the integral of the difference \(f-g\). If the curves cross, the interval is split at the crossing points.
The area between two curves is the integral of "top curve minus bottom curve": \(\int_a^b (f-g)\,dx\). The computation reduces to deciding which is on top and between which \(x\) values.
First find the crossings (solve \(f=g\)) to know the limits. If the curves cross in the middle, split the integral into pieces, or use absolute value.
Sketching before integrating avoids most sign errors.
Think about it
Find the area between \(y=x\) and \(y=x^2\) on [0,1].
\(\int_0^1 (x-x^2)\,dx=\frac12-\frac13=\frac16\).
Exercises 10
- Find the area between \(y=x\) and \(y=x^{2}\) on \([0,1]\).
- Find the area between \(y=2 x\) and \(y=x^{2}\) on \([0,2]\).
- Find the area between \(y=\sqrt{x}\) and \(y=x^{2}\) on \([0,1]\).
- Find the area between \(y=x + 2\) and \(y=x^{2}\) on \([-1,2]\).
- Find the area between \(y=x^{3}\) and \(y=0\) on \([0,1]\).
- If \(f\ge g\) on \([a,b]\), which integral gives the area between \(f\) and \(g\) on that interval?
- Find the area between \(y=\sin x\) and \(y=0\) on \([0,\pi]\).
- Find the area between \(y=x\) and \(y=x^{3}\) on \([0,1]\).
- Find the area between \(y=x^{2}\) and \(y=0\) on \([1,2]\).
- Find the area between \(y=x\) and \(y=2-x\) on \([0,2]\).
References
- James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.