integration

[katyt]

I'm having some trouble integrating this:

(Integral of) (sin(3pix))^2 from 0 to 1/3

I keep getting 0, but when I put it into my calculator, I get 1/6.
Could someone give me some help?

 

[soroban]

Hello, Katy!

Just a guess . . . you're probably integrating incorrectly.
. . It's tricky . . .

[INT] sin²(3 pi x) dx . . . from 0 to 1/3
.

To integrate sine-squared, we use: .sin²θ = (1/2) [1 - cos(2θ)]

. . So: .sin²(3πx) .= .(1/2) [1 - cos(6πx)]

We are integrating: .(1/2) [INT] [1 - cos(6πx)] dx

. . and we get: .(1/2) [x - (1/6π)sin(6πx) ]

Now plug in the limits . . .