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 (i) Parametric coordinates or open all working together.

(ii)

Parametric area

The line ll passes through the origin OO and AA. The region RR is bounded by CC and ll. Using an algebraic method, find the exact area of RR.

Explore this idea

Hint

  1. RR is the area under the curve from OO to AA, minus the triangle under ll.

  2. For the curve, write ∫y dx=∫ydxdt dt\int y\,dx=\int y\frac{dx}{dt}\,dt and change the limits to tt.

  3. Expect sin⁡2t\sin^2t, so use the double-angle formula.

Answer · (ii)

Area =1/6=1/6.

Explore the idea

Open interactive exploration

A sliver between the curve and its chord

The curve rises steeply, then flattens as it reaches A. The chord from O to A runs below it.

Number of stripsn = 4
Try an example

Where does one strip’s area come from?

1 of 4

The curves meet at this sample, so the strip has zero height and contributes zero. Its red marks use the left edge at x* = 0.

zero heightupper = lowerwidth Δx
Zoomed-in rectangle: width and height are scaled independently. The sample sets its upper and lower values across the whole strip.
Sample x*
0
Width Δx
0.1427
Upper value
0
Lower value
0
Height: upper − lower
0
Strip area
0

0.1427 ×(0 − 0)≈ 0

Add this contribution to those of the other 3 strips to get the sum below. Displayed values are rounded.

Strip width
0.143
0.571 ÷ 4
Sum of strips
0.14809
Exact, 1/6
0.16667
Error
−0.01858
11.15% too small

How the sum improves as n grows

  • Rising part of C, from O to A
  • Chord l from the origin to A
  • Rectangles of height (curve − line)
  • Exact area 1/6 ≈ 0.16667
Key idea

Parametric integration turns a curve we cannot write as y(x)y(x) into a simple trig integral, and the area comes out as the neat 16\frac16.

Ideas to try
  1. Look at the left end: the curve leaves O much more steeply than l. Why?
  2. Raise n to 40 and compare the strip total with 1/6.