Areas: Between a Curve and the Axes
Areas between a curve and the axes are found with a definite integral, taking the size of any part below the axis so the area is never negative.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Integral calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
The area between a curve and the \(x\)-axis is a definite integral. This Year 12 Mathematics Advanced topic (MAV-12-05) finds areas, taking the size of any part below the axis.
For \(f(x)\ge0\) on \([a,b]\), Area \(=\displaystyle\int_a^b f(x)\,dx\). Where the region is below the axis the integral is negative, so take its absolute value.
If the curve crosses the axis, split at the intercept and add the sizes of the pieces.
Method
- Integrate \(f\) between the limits.
- Take the size of any negative piece (below the axis).
- Add the pieces if the curve crosses the axis.
| \(\ \) | \(=\) | \(\big[\tfrac{x^{3}}{3}\big]_0^3=9\) |
| \(\ \) | \(=\) | \(\big[x^{2}\big]_0^4=16\) |
| \(\ \) | \(=\) | \(\big[\tfrac{x^{3}}{3}\big]_1^2=\tfrac73\) |
| \(\int_0^2(-x)dx\) | \(=\) | \(-2\) |
| \(\text{Area}\) | \(=\) | \(2\) |
Common pitfalls
Frequently asked questions
How do you find the area under a curve?
Integrate the function between the two x-values. For a curve above the axis, the definite integral is the area.
What if the curve is below the x-axis?
The integral will be negative, so take its absolute value to get the area.
What if the curve crosses the x-axis?
Split the interval at the intercept, find each piece's area separately, and add the sizes.
What is the area under y = x squared from 0 to 3?
It is the integral, x cubed over 3 evaluated from 0 to 3, which is 9.