# Homework Help: Average of sine function

1. Jan 24, 2016

### Dexter Neutron

• Member warned about posting with no effort
How to find the average value of sin2wt?

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Last edited by a moderator: Jan 24, 2016
2. Jan 24, 2016

### Krylov

How do you find the average of any periodic, integrable function?

3. Jan 24, 2016

### Dexter Neutron

4. Jan 24, 2016

### Krylov

Do you have any ideas yourself?

5. Jan 24, 2016

### Dexter Neutron

I know that average of any function is equal to sum of all the values of the function divided by number of values.
So we can integrate sin2wt is the interval 0 to 2π and divided it by the total number of values.
But what is the total number of values.How to find it?

6. Jan 24, 2016

### PeroK

First, note that $sin^2(wt)$ repeats every $\pi$ units.

What if you could find the average value of this function, call it $a$, and then you drew a rectangle of height $a$ from $0$ to $\pi$. What could you say about the area of that rectangle?

7. Jan 24, 2016

### Krylov

Indeed.
You correctly recalled the definition of the average of finitely many values $a_1,\ldots,a_n$ as
$$\frac{1}{n}\sum_{i=1}^n{a_i} \qquad (1)$$
Now, your problem is that for your function there is an infinitude of values. For such a case you need a new definition of "average". It is obtained by replacing the sum in (1) by an integral and dividing by the length of the interval. So, if $f : \mathbb{R} \to \mathbb{R}$ is a function, you can define its average over any interval $[a,b]$ as
$$\overline{f} := \frac{1}{b-a}\int_a^b{f(x)\,dx}$$
When the interval is unbounded, you have to use a limit. So, for your function $f(t) := \sin^2{\omega t}$ you get for its average over $\mathbb{R}$,
$$\overline{f} = \lim_{T \to \infty}\frac{1}{2T}\int_{-T}^T{f(t)\,dt}$$
However, because your function is periodic with period $\tau := \tfrac{\pi}{\omega}$, all you need to do to calculate the above is to integrate from $0$ to $\tau$ and divide by $\tau$ to obtain $\overline{f}$. (Hint: the answer does not depend on $\omega$.)

Last edited: Jan 24, 2016