# (definite integral) area between two trig functions

I'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever interval is in question, but am having trouble setting up the integral.

The two functions I'm dealing with are $y=5\cos(5x)$ & $y=5-5\cos(5x)$, the interval is $[0, \pi/5]$.

The tricky part for me (likely due to never having taken trig) is finding the point within the interval where the functions overlap (I know they do) to properly set up the integral.

Perhaps someone could explain to me (or direct me to an explanation) of how to find when two trig functions are equal.

Note that $$5\cos(5x)=5-5\cos(5x)\iff 10\cos(5x)=5\iff \cos(5x)=\frac{1}{2}.$$

• Thanks! I think I was just overthinking it. Jul 1, 2015 at 19:38

This gives us an intuition as to how to proceed, the blue curve is $5 \cos 5x$ and red is the other. The two curves intersect when $$5 \cos 5x = 5 - 5\cos 5x \iff \cos 5x = \frac{1}{2}$$

But $$\cos 5x = \frac{1}{2} \implies x = \frac{\pi}{15}$$ in the range we are concerned in.

So the area between the curves is $$\int_0^{\frac{\pi}{15}} 5 \cos 5x - (5 - 5\cos 5x) \, \mathrm{d}x + \int_{\frac{\pi}{15}}^\frac{\pi}{5} 5 - 5\cos 5x - 5 \cos 5x \, \mathrm{d}x$$