Answer
$$\int^{\pi/6}_0(1-\cos3t)\sin3tdt=\frac{1}{6}$$
$$\int^{\pi/3}_{\pi/6}(1-\cos3t)\sin3tdt=\frac{1}{2}$$
Work Step by Step
Part a) $$A=\int^{\pi/6}_0(1-\cos3t)\sin3tdt$$
We set $u=1-\cos3t$, which means $$du=-3(-\sin3t)dt=3\sin3tdt$$ $$\sin3tdt=\frac{1}{3}du$$
For $t=0$, we have $u=1-\cos0=1-1=0$
For $t=\pi/6$, we have $u=1-\cos(\pi/2)=1-0=1$
Therefore, $$A=\frac{1}{3}\int^1_0udu=\frac{1}{3}\times\frac{u^2}{2}\Big]^1_0=\frac{1}{6}\times u^2\Big]^1_0$$ $$A=\frac{1}{6}(1-0)=\frac{1}{6}$$
Part b) $$A=\int^{\pi/3}_{\pi/6}(1-\cos3t)\sin3tdt$$
We set up the same substitution as in part a).
For $t=\pi/3$, we have $u=1-\cos\pi=1-(-1)=2$
For $t=\pi/6$, we have $u=1-\cos(\pi/2)=1-0=1$
Therefore, $$A=\frac{1}{3}\int^2_1udu=\frac{1}{6}\times u^2\Big]^2_1$$ $$A=\frac{1}{6}(2^2-1^2)=\frac{1}{6}\times(3)=\frac{1}{2}$$
