Math Help - Squared sin and cos definite integrals question

1. Squared sin and cos definite integrals question

hi!

1. Is the definite integral over [sin(x)]^2 from 0 to a, is a/2? and also the cosine?

2. Is it because the average of the sin/cos functions is always 1/2?

3. and if so, the integration region doesn't matter because the function is positive?
I mean, what about integration from -a to a?

4. for example, from -3a to 0.5a? is it: 0.5*[3.5a]?

thanks!

2. $\displaystyle\int_0^a\sin^2xdx=\int_0^a\frac{1-\cos 2x}{2}dx$
Can you take it from here?

3. thanks for that!

but the thing with Int[sin^2] is 1/2 or something... what am I missing?

4. $\displaystyle\int_0^a\sin^2xdx=\int_0^a\frac{1-\cos 2x}{2}dx=\left.\frac{x}{2}\right|_0^a-\left.\frac{\sin 2x}{4}\right|_0^a=\frac{a}{2}-\frac{\sin 2a}{4}$
So, the result depends on $a$.