Integration - Understanding Results of Definite Integrals

1. Sketch the curve \(y = x(x - 3)(x - 5)\).
2. Find the following integrals.
   1. \(\displaystyle\displaystyle\int_{0}^{3} x^3 - 9x^2 + 18x \, \mathrm{d}x\)
   2. \(\displaystyle\displaystyle\int_{3}^{6} x^3 - 9x^2 + 18x \, \mathrm{d}x\)
   3. \(\displaystyle\displaystyle\int_{0}^{6} x^3 - 9x^2 + 18x \, \mathrm{d}x\)
3. Explain why the area between the curve and the \(x\)-axis in the interval \(0 \le x \le 6\) is not the answer to (b)(iii). And find the area between the curve and the \(x\)-axis for the interval \(0 \le x \le 6\).

In part (c), use the curve \(y=x^3-9x^2+18x=x(x-3)(x-6)\) from part (b). The sketch in part (a) is a separate curve.