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Question: What is the average value of a function \( \sin \left( X \right) \) on the interval \( \left[ {0,pi}...

What is the average value of a function sin⁡(X)\sin \left( X \right) on the interval [0,pi]\left[ {0,pi} \right] ?

Explanation

Solution

Hint : In order to determine the average value of the given function sin⁡(X)\sin \left( X \right) , first we need to simply integrate the function with the known formula of integration that is ∫sin⁡xdx=−cos⁡x+c\int {\sin xdx = - \cos x + c} . Then put the intervals given to us. Then for average divide the value obtained by π\pi and hence, the result is obtained.

Complete step-by-step answer :
We are given the function sin⁡(X)\sin \left( X \right) .
Since, we need to find the average value of the function so we would be integrating the value divided by π\pi to get that.
So, integrating sin⁡(X)\sin \left( X \right) and we get:
∫sin⁡(X)dx=−cos⁡(X)+c\int {\sin \left( X \right)dx = - \cos \left( X \right) + c}
Since, we are given the intervals so we would be excluding the constant part to get the exact results.
Applying the intervals for the function:
∫0πsin⁡(X)dx=−[cos⁡(X)]0π\int\limits_0^\pi {\sin \left( X \right)} dx = - \left[ {\cos \left( X \right)} \right]_0^\pi
Solving further by putting the upper and lower limits in the value obtained and we get:
∫0πsin⁡(X)dx=−[cos⁡(X)]0π =−[cos⁡(π)−cos⁡(0)] =−[−1−1] =−[−2] =2   \int\limits_0^\pi {\sin \left( X \right)} dx = - \left[ {\cos \left( X \right)} \right]_0^\pi \\\ = - \left[ {\cos \left( \pi \right) - \cos \left( 0 \right)} \right] \\\ = - \left[ { - 1 - 1} \right] \\\ = - \left[ { - 2} \right] \\\ = 2 \;
Since, we know Average value is the sum of value obtained divided by total number of terms. So, dividing the value obtained by π\pi .
Therefore, Average value of the function sin⁡(X)=∫0πsin⁡(X)dxπ=2π\sin \left( X \right) = \dfrac{{\int\limits_0^\pi {\sin \left( X \right)dx} }}{\pi } = \dfrac{2}{\pi } .
So, the correct answer is “ 2π\dfrac{2}{\pi } ”.

Note : Integration is basically used to find the areas, volumes, central point’s etc. But most often it is used to find the area under the graph of the function.
Similarly, all other values with different powers can be integrated using the power rule.
It’s very important to add a constant after integrating the equation because when we derive an equation their constant is removed.
But, when an interval is given that means the exact area could be found out, so no need to add a constant.