Question

I have a question

Answer 1

The red part which is equal to 1/2, please explain with calculus

Answer 2

@Luigi0210

Answer 3

@robtobey

Answer 4

@nincompoop

Answer 5

@Compassionate

Answer 6

@Preetha

Answer 7

i know same old question

Answer 8

but i still have doubt

Answer 9

@Somy

Answer 10

There is is mistake on the board in the image.
The second function shown is not \(\sin ^2\) but \(|\sin|\).

The mean value of \(|\sin|\) is not 1/2, but the mean value of \(\sin ^2\) is.

\(\sin ^2\) is a regular sine function between 0 and 1. Its mean value is clearly 1/2 is you plot it properly like this:
http://www.wolframalpha.com/input/?i=sin%28x%29^2

In the page by Wolfram, it is clear that the alternative form's average value is 1/2 since the cosine function's average is 0.

Answer 11

The second graph which is drawn on the board is the modulus of the sine function.which makes all values of the sine function positive.

Answer 12

yes, its 1/2(x-sin(x)cos(x))+C

but why did the professor take only 1/2?

Answer 13

@Vincent-Lyon.Fr

Answer 14

The average of a sine quaked function over a period is 0.5.
The green area is equal to the cyan area.



Think of some ground the surface of which is like the sine squared function and you are asked to level the ground.

Answer 15

Sorry. The spell checker converted my squared into quaked.

Answer 16

I got it.... :)
Thanks @Farcher

Answer 17

And Thanks @Vincent-Lyon.Fr