C2 2016 7 Edexcel

(a)

We have:

\begin{align*}\int \left(3x - x^{3/2}\right) \mathrm dx & = \frac 3 2 x^2 - \frac {x^{5/2}} {\frac 5 2} + C \\ & = \frac 3 2 x^2 - \frac 2 5 x^{5/2} + C\end{align*}

(b)

Note that y = 3x - x^{3/2} = x(3 - \sqrt x). So if y = 0 then either x = 0 or 3 = \sqrt x, ie. x = 9. So the area of S is equal to:

\begin{align*}\int_0^9 \left(3x - x^{3/2}\right) \mathrm dx & = \left[ \frac 3 2 x^2 - \frac 2 5 x^{5/2}\right]_0^9 \\ & = \frac 3 2 \times 9^2 - \frac 2 5 \times 9^{5/2} \\ & = \frac {243} {10}\end{align*}

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