finding a function to correlate frequency and energy

[Perdurat]

consider a function that describes a sine wave:
sin(x)
introducing a variable to modify the amplitude (A, default value=1):
A*sin(x)
introducing a variable to modify the frequency (F, default value=1):
A*sin(F*x)
for sin(x), between x=0 and x=pi (1/2 wavelength), the surface below the wave amounts to 2

how to find the function that describes the relationship between A and F such that the surface below 1/2 wavelength remains constant (=2) ?
thanks for your time

 

[MarkFL]

Hello, and welcome to FMH!

I would begin by writing:

[MATH]A\int_0^{\frac{\pi}{\omega}} \sin(\omega x)\,dx=2 [/MATH]
Use the substitution:

[MATH]u=\omega x\implies du=\omega\,dx[/MATH]
And we have:

[MATH]\int_0^{\pi} \sin(u)\,du=\frac{2\omega}{A}[/MATH]
Evaluating the definite integral, we get:

[MATH]2=\frac{2\omega}{A}[/MATH]
And this implies:

[MATH]A=\omega[/MATH]

 

[Perdurat]

thanks for your reply
have done some maths with integrals, 30 years ago, it's a bit rusty...
mostly rust....