What's the value of the definite integral $\int_{0}^{20π} |\sin(x)| dx$?

Question

I'm not sure about the answer to this, I tried using the calculator to find the answer, it gave me approximately $12.6$, when working it out by hand, the answer was $0$, what's the answer? And how to calculate it right?

$$\int_{0}^{20π} |\sin(x)| dx$$

Answer 1

I'm assuming that $ | \ | $ refers to the absolute value.

Notice that $\int_0^{2 \pi} |sin(x)| dx = 2 \int_0^{\pi} sin(x) dx = 4.$ (The first equality is obvious if you draw the graph.)

You can do the same for the interval $[2\pi, 4\pi]$, $[4\pi, 6\pi]$ etc. So to get the whole interval $[0, 20\pi]$ you will have to do it 10 times which gives:

$\int_0^{20\pi} |sin(x)| dx = 40$.

Answer 2

Break up the domain where $sin(x)<0$ and $sin(x)>0$.

$$\int_0^{20\pi}|sin(x)|dx=\int_0^{\pi}sin(x)dx-\int_\pi^{2\pi}sin(x)dx+\cdots+\int_{18\pi}^{19\pi}sin(x)dx-\int_{19\pi}^{20\pi}sin(x)dx$$

This is because $sin(x)<0$ when $x\in (k\pi,[k+1]\pi)$ and $k$ is odd.
Likewise, $sin(x)>0$ when $x\in (k\pi,[k+1]\pi)$ and $k$ is even.

So if $sin(x)<0$ then $|sin(x)|=-sin(x)>0$

Answer 3

Note that $$\int_0^{20\pi} |\sin(x)|dx = 20\int_0^\pi\sin(x)dx$$ So then the question boils down to $$\int_0^\pi \sin(x)dx = -\cos(\pi)+\cos(0) = 2 \implies \int_0^{20\pi} |\sin(x)|dx =40 $$