Areas Involving Trigonometric Functions
Learn areas involving trigonometric functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). Sine and cosine curves rise above and dip below the \(x\)-axis in regular arches, so their areas need integration in radians and careful attention to sign.
You will learn to integrate \(\sin x\) and \(\cos x\) over exact intervals, split regions where a curve crosses the axis, add the separate parts as positive areas and use symmetry to shorten the working — skills used whenever a wave or cycle is measured.
Theory
Areas under trig curves use the trig integral rules (radians). This Year 12 Mathematics Advanced topic (MAV-12-05) evaluates areas under \(\sin\) and \(\cos\).
Area \(=\displaystyle\int_a^b f(x)\,dx\) for \(f\ge0\). \(\sin x\ge0\) on \([0,\pi]\), so its area there is a clean integral. Where the curve dips below the axis, take the size of that part.
Note: areas enclosed between two curves are studied in Mathematics Extension 1, not in this course.
Method
- Integrate the trig function.
- Substitute the limits (radians).
- Take the size of any part below the axis.
| \(\ \) | \(=\) | \(\big[-\cos x\big]_0^{\pi}=2\) |
| \(\ \) | \(=\) | \(\big[\sin x\big]_0^{\pi/2}=1\) |
| \(\ \) | \(=\) | \(\big[-\cos x\big]_0^{\pi/2}=1\) |
| \(\ \) | \(=\) | \(2\) |
Common pitfalls
Frequently asked questions
What is the area under sin x from 0 to pi?
It is 2, from minus cos x evaluated between 0 and pi.
What is the area under cos x from 0 to pi over 2?
It is 1, from sin x evaluated between 0 and pi over 2.
Why must the angles be in radians?
The trig integral rules only hold when the angle is measured in radians.
What if the sine curve goes below the axis?
Over that part the integral is negative, so take its absolute value for the area.