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# find the average value of the function g(x)=2sin(x)+e^(x/pi) on the interval 0, (2pi)

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find the average value of the function g(x)=2sin(x)+e^(x/pi) on the interval 0, (2pi)

Nov 30, 2019

#1
+106954
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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.

Dec 1, 2019
#2
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Could you solve with u-substitution?

Dec 2, 2019
#3
+106954
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idk  Can you give me more info on what you want the u substitution to be?

Melody  Dec 2, 2019
#4
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x/pi

Guest Dec 3, 2019
#5
+106954
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Your a guest. Indistinguishable from all other guests.

If you want to say something you need to say it properly.

Melody  Dec 3, 2019