Areas Involving Trigonometric Functions
Areas involving trigonometric functions use definite integrals of \(\sin x\) and \(\cos x\) to find regions under and between trig curves (radians).
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
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.
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.