Keep the parameter branch when eliminating t

x=t-\sin t

Total marks: 10

Solution

(i)

Parametric coordinates

The curve $C$ has parametric equations $x=t-\sin t$ , $y=\sin^2t$ for $0\le t\le\pi$ . The point $A$ has parameter $t=\frac\pi2$ and is a maximum point. Find the coordinates of $A$ .

Hint

Substitute $t=\frac\pi2$ into both equations.

Answer · (i)

A=(\pi/2-1,1)

(ii)

Parametric area

The line $l$ passes through the origin $O$ and $A$ . The region $R$ is bounded by $C$ and $l$ . Using an algebraic method, find the exact area of $R$ .

Explore this idea

Hint

$R$ is the area under the curve from $O$ to $A$ , minus the triangle under $l$ .
For the curve, write $\int y\,dx=\int y\frac{dx}{dt}\,dt$ and change the limits to $t$ .
Expect $\sin^2t$ , so use the double-angle formula.

Answer · (ii)

=1/6

Explore the idea

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.

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

The vertical scale fits n ≥ 3; earlier values may lie outside the view.

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 $y(x)$ into a simple trig integral, and the area comes out as the neat $\frac16$ .

Ideas to try

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

(iii)

Eliminating the parameter

Show that the Cartesian equation of $C$ is $x=\sin^{-1}\sqrt y-\sqrt y$ .

Hint

Express $\sin t$ in terms of $y$ , then $t$ in terms of $y$ , and substitute into $x$ .

Answer · (iii)

x=\arcsin\sqrt y-\sqrt y

(iv)

Volume of revolution

Find the volume generated when $R$ is rotated through $2\pi$ radians about the $y$ -axis.

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Hint

Spinning $R$ about the $y$ -axis, slice horizontally. That is why you needed $x$ in terms of $y$ .
The outer boundary is the line $l$ (a cone), the inner boundary is the curve.
Volume = cone − (volume swept by the area between the curve and the $y$ -axis).

Answer · (iv)

\approx0.253

Explore the idea

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.

Keep the parameter branch when eliminating t

Solution

Parametric coordinates

Substitute

Parametric area

Split the area

Change to $t$

Split and integrate

Subtract the triangle

A sliver between the curve and its chord

Where does one strip’s area come from?

Eliminating the parameter

Solve for $\sin t$

Solve for $t$

Substitute into $x=t-\sin t$

Volume of revolution

The cone from the line

Subtract the volume inside the curve

Evaluate by calculator

A cone with a curved bowl taken out

Substitute

Split the area

Change to ttt

Split and integrate

Subtract the triangle

Solve for sin⁡t\sin tsint

Solve for ttt

Substitute into x=t−sin⁡tx=t-\sin tx=t−sint

The cone from the line

Subtract the volume inside the curve

Evaluate by calculator

Change to $t$

Solve for $\sin t$

Solve for $t$

Substitute into $x=t-\sin t$