MHB Due to a symmetry of the cosine we can just double the integral from 0 to 1

[tmt1]

Hello,

∫|cos(px/2)|dx between [0,2]

I encountered this rule. How does this apply to other intervals of say [3,4],[7,9] etc.

Are the numbers both halved?

so [3,4] becomes [1.5,2] etc?

Also, does this rule apply to all symmetrical functions?

Thank you,

Tim

 

[zzephod]

Hello,

∫|cos(px/2)|dx between [0,2]

I encountered this rule. How does this apply to other intervals of say [3,4],[7,9] etc.

Are the numbers both halved?

so [3,4] becomes [1.5,2] etc?

Also, does this rule apply to all symmetrical functions?

Thank you,

Tim

Assuming you mean:$$ I=\int_0^2 |\cos(\pi x/2)| \;dx = 2\int_0^1 |\cos(\pi x/2)| \;dx $$.

You can do this because the integrand is symmetric about the mid-point of the interval $$(0,2)$$. But also the integrand is periodic with period $$2$$, which means that its integral over any interval of a integer number of periods is equal to its integral over any other interval of the same length. That is, for any real $$a$$ :

$$\phantom{xxxxxxx} \int_0^2 |\cos(\pi x/2)| \;dx=\int_a^{a+2} |\cos(\pi x/2)| \;dx$$

.