Keep the parameter branch when eliminating t

Parametric curve C and line l enclosing R; O, A and R suppliedSource-given figure redrawn with D3. Parametric curve C and line l enclosing R; O, A and R supplied. Only labels supplied in the original question are included.xyOARl
Question diagram 1
Parametric curve C and line l enclosing R; O, A and R suppliedSource-given figure redrawn with D3. Parametric curve C and line l enclosing R; O, A and R supplied. Only labels supplied in the original question are included.xyOARl

Given parametric curve C and line l through O and A, enclosing R.

C: x=t−sin⁡tx=t-\sin t, y=sin⁡2ty=\sin^2t, 0≤t≤π0\le t\le\pi. A has parameter π/2\pi/2 and is a maximum point. Line l passes through the origin and A. R is bounded by C and l as shown.

(i) Find A. mark

(ii) Using an algebraic method, find the exact area of R. marks

(iii) Show the Cartesian equation of C is x=sin⁡−1y−yx=\sin^{-1}\sqrt y-\sqrt y. mark

(iv) Find the volume when R is rotated 2π2\pi radians about the yy-axis. marks

[2023/ACJC/Prelim/I/8]

Total marks: 10

Solution

If this part uses an earlier result, revisit (ii) Parametric area or open all working together.

(iii)

Eliminating the parameter

Show that the Cartesian equation of CC is x=sin⁡−1y−yx=\sin^{-1}\sqrt y-\sqrt y.

Hint

  1. Express sin⁡t\sin t in terms of yy, then tt in terms of yy, and substitute into xx.

Answer · (iii)

x=arcsin⁡y−yx=\arcsin\sqrt y-\sqrt y on the rising arc.

Valid 0≤t≤π/2 only. On π/2≤t≤π use x=π−arcsin√y−√y.