Originally Posted by

**Archie Meade** That substitution gives you a $\displaystyle sint$ to contend with.

Instead.. $\displaystyle 2sin^2\left(\frac{t}{2}\right)=1-cost$

Then you can use $\displaystyle u=\frac{t}{2}$

This graph does not cross the x-axis as the maximum value of $\displaystyle cost$ is 1.

The notation you asked about....

That graph crosses the x-axis every 90 degrees,

however it is symmetrical about the x-axis.

As you are integrating from 0 to 360 degrees,

we calculate 4 times the integral from 0 to 90 degrees.

$\displaystyle sintcost=0\ for\ t=0^o,\ 90^0,\ 180^o,\ 270^o,\ 360^o$

Also we have negative f(t) for $\displaystyle \frac{{\pi}}{2}<t<{\pi}$ and $\displaystyle \frac{3{\pi}}{2}<t<2{\pi}$

$\displaystyle f(-t)=-f(t)$