Q. 55 - Calculus

Question

For each function f and interval [a, b] in Exercises 53–55, approximate the average value of f from x = a to x = b, using a sample size of at least eight. 

$f\left(x\right)=\sin x, \left[0,2\mathrm{\pi }\right]$

Step-by-Step Solution

Answer

The average value of f from x = a to x = b, using a sample size of at least eight is $0.$

Step 1. Given Information.

The given function and the interval is $f\left(x\right)=\sin x, \left[0,2\mathrm{\pi }\right].$

Step 2. Approximating the average value.

To approximate the average value of f from x = a to x = b, using a sample size of at least eight, we will use the formula: $\left(f\left(x_0\right)+......f\left(x_n\right)\right)△x, △x=\frac{b-a}{n}.$

Thus,

$$\begin{gathered} △x=\frac{2\mathrm{\pi }-0}{8} \\ △x=\frac{\mathrm{\pi }}{4} \end{gathered}$$

Now,

$$\begin{gathered} =\left(f\left(\frac{\mathrm{\pi }}{4}\right)+f\left(\frac{\mathrm{\pi }}{2}\right)+f\left(\frac{3\mathrm{\pi }}{4}\right)+f\left(\mathrm{\pi }\right)+f\left(\frac{5\mathrm{\pi }}{4}\right)+f\left(\frac{3\mathrm{\pi }}{2}\right)+f\left(\frac{7\mathrm{\pi }}{4}\right)+f\left(2\mathrm{\pi }\right)\right) \\ \left(\frac{\mathrm{\pi }}{4}\right) \\ =(\sin \left(\frac{\mathrm{\pi }}{4}\right)+\sin \left(\frac{\mathrm{\pi }}{2}\right)+\sin \left(\frac{3\mathrm{\pi }}{4}\right)+\sin \left(\mathrm{\pi }\right)+\sin \left(\frac{5\mathrm{\pi }}{4}\right)+\sin \left(\frac{3\mathrm{\pi }}{2}\right) \\ +\sin \left(\frac{7\mathrm{\pi }}{4}\right)+\sin \left(2\mathrm{\pi }\right))\left(\frac{\mathrm{\pi }}{4}\right) \\ =0 \end{gathered}$$

Therefore, the average value is $0.$