Average value of sinx?

Question

How do you find the average value of sinx between two values of x. ie the average value of sinx between x=0 and x=pi/3

Answer 1

The average value of a function is the integral of the function divided by the length of the interval over which you are integrating.
Thus:
3/ pi Integral ( sinx) dx, 0, pi/3

Answer 2

Average value = area under the curve y = sin x between x = 0 and x = pi/3 divided by the length of the interval, i.e., pi/3.

Thus, average value = (3/pi) integral ( x=0 to x=pi/3) sin x dx
=(3/pi) ( - cos x ) ( x =0 to x=pi/3)
=(3/pi) [ - cos(pi/3) + cos 0 ]
= (3/pi) ( - 1/2 + 1)
= 3/(2pi)

Answer 3

What is an average?
How accurate do we want it?

Take two values: sin(0), sin(pi/3), add them up and divide by 2 (very rough estimate).
Take 10 intervals (11 values):
sin(0), sin(pi/30), sin(2pi/30),..., sin(10pi/30)
add them up, divide by 11 (much better estimate).

The limit will be when you add up an infinite number of values and divide it by the infinite number of measurements: the integral.

Take the definite integral from 0 to pi/3 (this gives you the area under the curve) and divide by the width of the area (pi/3).

Answer 4

I don't think it can be done as the values of sinx follow a curve. It could be done with a straight line graph by taking the middle value, but if you were to do this here (ie. when x=pi/6) it wouldnt be right. (hmm...I dont know if u can do it & I'm just being dumb!)