2019 NJC P2 Q4

If \(k\) is a non-zero integer, find \(\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\sin kx\,\mathrm{d}x}\) and \(\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\cos kx\,\mathrm{d}x}\).[2]

Let \(m\) and \(n\) be positive integers.

1. Show that
   \(\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\sin \left( mx \right)\sin \left( nx \right)\mathrm{d}x=\left\{ \begin{aligned} \pi \mathrm{ if }m&=n, \\ 0\mathrm{ if }\,m&\ne n. \\ \end{aligned} \right.}\)[4]
2. By considering the formula for \(\cos \left( mx+nx \right)\), evaluate
   \(\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\cos \left( mx \right)\cos \left( nx \right)\mathrm{d}x}\).

Suppose \(a\) is a positive integer.

3. Show that \(\displaystyle\displaystyle \displaystyle\int_{-\pi }^{\pi }{\left| \sin \left( ax \right) \right|\mathrm{d}x}=4\).[2]
4. Hence, find the exact volume of the solid formed when the region bounded by the curve \(y=\left| \sin \left( ax \right) \right|+1\), the \(x\)-axis and the lines \(x=-\pi \) and \(x=\pi \), is rotated completely about the \(x\)-axis.[3]