find the average value of the function g(x)=2sin(x)+e^(x/pi) on the interval 0, (2pi)
find the average value of the function g(x)=2sin(x)+e^(x/pi) on the interval 0, (2pi)
The average is the infinite sum of the g(x) values divided by the number of x values.
The integral sign is an stylized S because it stand for Sum.
I know this is not explained well. There would be a number of youtube clips on it if you look.
anyway....
Average g(x) value =
\(=\frac{1}{2\pi-0}*\displaystyle\int_0^{2\pi}\;2sin(x)+e^{x/\pi}\;dx \\ =\frac{1}{2\pi}*\left[-2cos(x)+ \pi e^{x/\pi}\right]_0^{2\pi}\\ =\frac{1}{2\pi}*\left [(-2cos(2\pi)+ \pi e^{2\pi/\pi})-(-2cos(0)+ \pi e^{0/\pi}) \right]\\ =\frac{1}{2\pi}*\left [(-2+ \pi e^2)-(-2+ \pi ) \right]\\ =\frac{1}{2\pi}*\left [(-2+ \pi e^2+2- \pi ) \right]\\ =\frac{1}{2}*\left [( e^2- 1) \right]\\ =\frac{e^2-1}{2} \)
Here is a visual representation of the average value of the function.
The average value is the height of the horizonal orange line.