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 (iii) Eliminating the parameter or open all working together.

(iv)

Volume of revolution

Find the volume generated when RR is rotated through 2π2\pi radians about the yy-axis.

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Hint

  1. Spinning RR about the yy-axis, slice horizontally. That is why you needed xx in terms of yy.

  2. The outer boundary is the line ll (a cone), the inner boundary is the curve.

  3. Volume = cone − (volume swept by the area between the curve and the yy-axis).

Answer · (iv)

Volume ≈0.253\approx0.253 cubic units (3 s.f.).

Rotate the source’s O-to-A region using the rising branch only.

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A cone with a curved bowl taken out

The line makes a cone. The curve, which bulges to its left, removes a thin curved shell from inside it.

Open the interactive view to explore this geometry in 3D.