1999 Paper 3 Q6

Year: 1999
Paper: 3
Question Number: 6

Course: LFM Pure and Mechanics
Section: Parametric equations

Difficulty: 1700.0 Banger: 1516.0

parametric equations curve sketching area enclosed by curve trigonometric parametrisation astroid integration Wallis integral closed curve

Problem

A closed curve is given by the equation $$ x^{2/n} + y^{2/n} = a^{2/n} \eqno(*) $$ where $n$ is an odd integer and $a$ is a positive constant. Find a parametrization $x=x(t)$, $y=y(t)$ which describes the curve anticlockwise as $t$ ranges from $0$ to $2\pi$. Sketch the curve in the case $n=3$, justifying the main features of your sketch. The area $A$ enclosed by such a curve is given by the formula $$ A= {1\over 2} \int_0^{2\pi} \left[ x(t) {\d y(t)\over \d t} - y(t) {\d x(t)\over \d t} \right] \,\d t \,. $$ Use this result to find the area enclosed by ($*$) for $n=3$.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

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Banger Rating: 1516.0

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Problem source

A closed curve is given by the equation
$$
x^{2/n} + y^{2/n} = a^{2/n} \eqno(*)
$$
where $n$ is an odd integer and $a$ is a positive constant.
Find a parametrization $x=x(t)$, $y=y(t)$ which
describes the curve  anticlockwise as $t$ ranges from $0$ to $2\pi$. 
Sketch the curve in the case $n=3$, justifying the main features 
of your sketch.
The area $A$ enclosed by such a curve
is given by the formula
$$
A= {1\over 2} \int_0^{2\pi} \left[ x(t) {\d y(t)\over \d t} -
y(t) {\d x(t)\over \d t} \right] \,\d t \,.
$$
Use this result to find the area enclosed by ($*$) for $n=3$.

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