The curve \(C\) has equation \(y=f(x)\), where \(f(x)=x\mathrm e^{-x^2}\).
Sketch the curve C.[2]
Find the exact coordinates of the turning points on the curve.[4]
Use the substitution \(u=x^2\) to find \(\int_0^n f(x)\,\mathrm dx\), for \(n>0\). Hence find the area of the region between the curve and the positive \(x\)-axis.[4]
Find the exact value of \(\int_{-2}^2|f(x)|\,\mathrm dx\).[2]
Find the volume of revolution when the region bounded by the curve, the lines \(x=0\), \(x=1\) and the \(x\)-axis is rotated completely about the \(x\)-axis. Give your answer correct to 3 significant figures.[2]