Signed integral of an odd function and the true area

A function \(\mathrm f\) is odd: \(\mathrm f(-x)=-\mathrm f(x)\) for all \(x\).

1. Using the symmetry of the graph of \(y=\mathrm f(x)\), explain why \(\displaystyle\int_{-a}^{a}\mathrm f(x)\,\mathrm dx=0\) for every \(a>0\).[2]
2. Hence write down the value of \(\displaystyle\int_{-3}^{3}\left(x^3-9x+2\right)\mathrm dx\), showing your reasoning.[2]
3. A student says that the area enclosed by the curve \(y=x^3-9x\) and the \(x\)-axis is \(\displaystyle\int_{-3}^{3}\left(x^3-9x\right)\mathrm dx=0\). Explain the error and find the correct area.[4]