A2 October 2021 Paper 1 Q6

OCR ACurrent spec3 marksIntegration

6 \(O\) is the origin of a coordinate system whose units are cm.
The points \(A\), \(B\), \(C\) and \(D\) have coordinates \((1, 0)\), \((1, 4)\), \((6, 9)\) and \((0, 9)\) respectively.
The arc \(BC\) is part of the curve with equation \(x^2 + (y - 10)^2 = 37\).
The closed shape \(OABCD\) is formed, in turn, from the line segments \(OA\) and \(AB\), the arc \(BC\) and the line segments \(CD\) and \(DO\) (see diagram).
A funnel can be modelled by rotating \(OABCD\) by \(2\pi\) radians about the \(y\)-axis.

Grid with x from -10 to 10 and y from 0 to 10: the shaded shape OABCD with O at the origin, A(1, 0), B(1, 4), an arc curving up from B to C(6, 9), and D(0, 9) on the y-axis

Find the volume of the funnel according to the model. [3]

Mark scheme

Scheme	Marks	AO
For \(AB\), \(V = \pi \times 1^2 \times 4 = 12.566\ldots\)
For \(BC\), \(\displaystyle V = \int_a^b \pi x^2\,\mathrm{d}y = \pi\int_4^9 \left(37 - (y - 10)^2\right)\mathrm{d}y\)	M1	3.3
\(= 356.05\ldots\)	A1	1.1
\(\Rightarrow\) Total \(V = 356.05\ldots + 12.566\ldots = 368.61\ldots\) \(= 369\ \left(\mathrm{cm}^3\right)\) to 3 sf	A1	3.4
	[3]

Notes

For \(AB\): \(4\pi\)

M1: Split into two parts and use formulae
An integral and an attempt at the volume of a cylinder must be seen

A1: Integration – ignore limits BC
\(\frac{340}{3}\pi\)

A1: Units are not required
\(\frac{352}{3}\pi\)