How to Calculate the Average Value of a Sinusoid Over a Given Interval?

[perryben]

If I had a sinusoid, how would I find the average value of it over a given interval. Say -pi/5 to pi/5 for instance. Thanks everybody.

 

[benorin]

The average value of a function, say $$f(x),$$ over the interval [a,b] is given by the the formula

$$f_{\mbox{ave}}=\frac{1}{b-a}\int_{x=a}^{b}f(x)dx$$

where I have assumed that $$f(x),$$ is properly integrable over [a,b].

 

[HallsofIvy]

The point is: if you had a constant function, f(x)= c, the "area under the curve" from a to b would f(x)(b-a)= c(b-a). With a variable function, that area is $$\int_a^b f(x)dx$$.

If f_ave is the average of the function we must have
$$\int_a^b f(x)dx= f_{ave}(b-a)$$

 

[mathman]

If I had a sinusoid, how would I find the average value of it over a given interval. Say -pi/5 to pi/5 for instance. Thanks everybody.

The sine function is odd. Therefore the average over an interval symmetric around 0 will be 0.