Area under and between trigonometric curves
The trigonometric integrals can be used to find the areas under and between curves as we have done with other functions.
It is important to sketch the curve, when calculating the area between it and an axis using integrals, to ensure any regions below the $x$x-axis are dealt with appropriately. If there is a region below the $x$x-axis the absolute value of the integral must be used in the area calculation.
To find the shaded area in the diagram above we could split the integral as follows:
$\text{Area}=\int_0^{\pi}f\left(x\right)\ dx+\left|\int_{\pi}^{2\pi}f\left(x\right)\ dx\right|$Area=∫π0f(x) dx+|∫2ππf(x) dx|
Or we can make use of the symmetry in this case:
$\text{Area}=2\times\int_0^{\pi}f\left(x\right)\ dx$Area=2×∫π0f(x) dx (since the areas are congruent)
Practice questions
Question 1
Find the exact area of the shaded region enclosed by the curve $y=\sin2x$y=sin2x, the $x$x-axis and the lines $x=\frac{\pi}{4}$x=π4 and $x=\frac{3\pi}{4}$x=3π4.
Question 2
Find the area enclosed by the curves $y=\sin x$y=sinx and $y=\cos x$y=cosx, and the lines $x=0$x=0 and $x=\frac{\pi}{4}$x=π4.