AS June 2022 Paper 1 Q10

AQACurrent spec6 marksIntegration

10 The diagram below shows an ellipse \(E\)

The coordinate axes are the lines of symmetry of \(E\)

Ellipse E centred on O, crossing the x-axis at −3 and 3 and the y-axis at 2 and −2

(a) Write down an equation of \(E\) [2 marks]

(b) The region bounded by the \(x\)-axis and the ellipse \(E\) for \(y \geqslant 0\) is shaded in the diagram below.

The ellipse E with the region above the x-axis, between x = −3 and x = 3, shaded

A solid \(S\) is formed by rotating the shaded region through \(360^\circ\) about the \(x\)-axis.

Show that the volume of \(S\) is \(a\pi\) where \(a\) is an integer to be found. [4 marks]

Mark scheme (a)

Scheme	Marks	AO
Writes any equation of a non-circular ellipse.	M1	1.1a
Obtains a correct equation of E. Accept \(3^2\) for 9 and \(2^2\) for 4	A1	1.1b
	(2)

Typical solution

\[\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1\]\[\frac{x^2}{9} + \frac{y^2}{4} = 1\]

Mark scheme (b)

Scheme	Marks	AO
Obtains a correct expression for \(y^2\) (or \(y\)) in terms of \(x\) FT their ellipse equation.	B1F	1.1b
Uses the formula for volume of revolution to write any expression of the form \(\int (mx^2 + c)\) for any non-zero \(c\) Condone missing \(\pi\), \(\mathrm{d}x\) and missing or incorrect limits. Condone \(\int (my^2 + c)\)	M1	3.1a
Writes a fully correct expression for the volume, eg \(\displaystyle\pi\int_{-3}^{3}\left(4 - \frac{4x^2}{9}\right)\mathrm{d}x\) Condone missing brackets.	R1	2.1
Obtains the correct volume in exact form.	A1	1.1b
	(4)
	(6 marks)

Typical solution

\[\frac{y^2}{4} = 1 - \frac{x^2}{9}\]\[y^2 = 4 - \frac{4x^2}{9}\]\[\begin{aligned}\text{Volume} &= \pi\int_{-3}^{3}\left(4 - \frac{4x^2}{9}\right)\mathrm{d}x \\ &= \pi\left[4x - \frac{4x^3}{27}\right]_{-3}^{3} \\ &= \pi\left(4 \times 3 - \frac{4 \times 3^3}{27}\right) - \pi\left(4 \times (-3) - \frac{4 \times (-3)^3}{27}\right) \\ &= \pi(12 - 4) - \pi(-12 + 4) \\ &= 16\pi\end{aligned}\]