Differentiation and applications

Question code: bb3

Question 0605

A curve has parametric equations

x = \sin^2 t, \quad y = \cos^3 t, \quad \textrm{for } 0 \leq t \leq {\textstyle \frac{\pi}{2}}.

Sketch the curve

The normal to the curve at the point

{( \sin^2 \theta, \cos^3 \theta ),}

where

{0 < \theta < \frac{1}{2} \pi,}

meets the

{x}

- and

{y \textrm{-axes }}

{P}

and

{Q}

respectively. The origin is denoted by

{O.}

Show that the area of triangle

{OPQ}

\frac{1}{ 12 \cos \theta } \Big| 3 \cos^{4} \theta - 2 \sin^2 \theta \Big|^2.

Show that the area under the curve for

{0 \leq t \leq \frac{\pi}{2}}

2 \int_0^{\frac{\pi}{2}} \sin t \cos^{4} t \, \mathrm{d} t,

and use the substitution

{u = \cos t}

to find this area.

Answer

{\frac{2}{5} \textrm{ units}^2.}

Question code: bb3