Find the average value of the function
Over the indicated interval .

Question

Find the average value of the function ​  ​
Over the indicated interval  .
​

Quizplus · Accepted Answer

To find the average value of a function over an interval, we use the formula: 
Average value = (1/b-a) * Integral from a to b of f(x) dx

In this case, the interval is from 0 to 2pi, so a=0 and b=2pi.

Plugging in the given function, we get:

(1/2pi-0) * Integral from 0 to 2pi of (cos^2(x/2) - sin^2(x/2)) dx

Using the identity cos(2theta) = cos^2(theta) - sin^2(theta), we can simplify to:

(1/2pi-0) * Integral from 0 to 2pi of (1/2)*(cos(x) + 1) dx

Evaluating the integral, we get:

(1/2pi) * [sin(x) + x] from 0 to 2pi

Plugging in the limits, we get:

(1/2pi) * [(sin(2pi) + 2pi) - (sin(0) + 0)]

Simplifying, we get:

(1/2pi) * (2pi) = 1

Therefore, the average value of the function over the interval 0 to 2pi is 1.

The best answer choice is A.

Find the Average Value of the Function ​ ​