FP3 June 2011 Q1
1. The curve \(C\) has equation \(y = 2x^3,\ 0 \leqslant x \leqslant 2\).
The curve \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis.
Using calculus, find the area of the surface generated, giving your answer to 3 significant figures. (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6x^2\) and so surface area \(= 2\pi\displaystyle\int 2x^3\sqrt{(1 + (6x^2)^2)}\,\mathrm{d}x\) | B1 |
| \(= 4\pi\left[\dfrac{2}{3 \times 36 \times 4}(1 + 36x^4)^{\frac{3}{2}}\right]\) | M1 A1 |
| Use limits 2 and 0 to give \(\dfrac{4\pi}{216}\left[13860.016 - 1\right] = 806\) (to 3 sf) | DM1 A1 |
| (5) | |
| (5 marks) |
Notes
B1 Both bits CAO but condone lack of \(2\pi\)
1M1 Integrating \(\displaystyle\int\left(y\sqrt{1 + \left(\text{their }\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2}\right)\mathrm{d}x\), getting \(k(1 + 36x^4)^{\frac{3}{2}}\), condone lack of \(2\pi\)
If they use a substitution it must be a complete method.
1A1 CAO
2DM1 Correct use of 2 and 0 as limits
2A1 CAO