QCE Specialist Maths: Integration and applications of integration Q18 (2025, Paper 2)

Integration and applications of integration — Question 18

QCAA 2025, Paper 2 · 7 marks

Q18 · 2025 · Technology-activeComplex unfamiliar7 marks

QUESTION 18 (7 marks)

Polar curves are defined by points that are a variable distance of

r

units from the origin and dependent on the angle

\theta

(in radians) measured from the positive

x

-axis. Consider the polar curve

r=1+\cos(\theta)

. A table of four polar coordinates on this curve is shown.

$\theta$	$r$
$0$	$2$
$\pi/6$	$1+\sqrt3/2$
$\pi/3$	$1.5$
$\pi/2$	$1$

The graph shows the polar curve

r=1+\cos(\theta)

for

0\le\theta\le2\pi

on a Cartesian plane. The polar coordinates from the table have been plotted on the curve.

The length of a polar curve,

L

, from

\theta=a

\theta=b

can be determined using the rule

L=\int_a^b\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta.

Use a complete algebraic method to determine the length of the section of the given polar curve that lies above the

x

-axis.

Cartesian plot of the polar curve r equals 1 plus cos theta with four plotted polar-coordinate points.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

L=4

units.

Worked solution

The section above the

x

-axis corresponds to

0\le\theta\le\pi

. Since

r=1+\cos\theta,\qquad\frac{dr}{d\theta}=-\sin\theta,

L=\int_0^\pi\sqrt{(1+\cos\theta)^2+(-\sin\theta)^2}\,d\theta.

Expand and use

\sin^2\theta+\cos^2\theta=1

L=\int_0^\pi\sqrt{2+2\cos\theta}\,d\theta.

Using

1+\cos\theta=2\cos^2(\theta/2)

L=\int_0^\pi2\cos\left(\frac\theta2\right)d\theta,

where the cosine is non-negative on the interval. Thus

L=4\sin\left(\frac\theta2\right)\Big|_0^\pi=4\ \text{units}.

correctly determines the values of

a

and

b

by considering the curve length above the

x

-axis

[1 mark]

correctly determines

dr/d\theta

[1 mark]

determines an expression for the integrand in expanded form

[1 mark]

uses a suitable identity to determine a simplified integrand

[1 mark]

uses a suitable double-angle identity within the integrand

[1 mark]

uses a suitable integration method to determine an expression for

L

[1 mark]

determines the required length

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

View the full QCAA marking guide

How many marks did you earn?

Compare your working with the guide above.

Integration and applications of integration — Question 18

QUESTION 18 (7 marks)

QCAA guide · typeset solution

Related questions