Use yy as the coordinate that only goes one way

Parametric curve C and line l with named intersections P and QSource-given figure redrawn with D3. Parametric curve C and line l with named intersections P and Q. Only labels supplied in the original question are included.xyOPQCl
Question diagram 1
Parametric curve C and line l with named intersections P and QSource-given figure redrawn with D3. Parametric curve C and line l with named intersections P and Q. Only labels supplied in the original question are included.xyOPQCl

Given parametric curve C and line l with intersections P and Q.

C: x=3t2x=3t^2, y=t3+3ty=t^3+3t, where tt is real. A line l cuts C at two points P and Q with parameters t=1t=1 and t=−1/2t=-1/2, respectively, as shown. Find the area bounded by C and l to 3 significant figures. marks

[2011/TPJC/Prelim/I/1]

Solution

A

Parametric area

The curve CC has parametric equations x=3t2x=3t^2, y=t3+3ty=t^3+3t, where tt is real. A line ll cuts CC at the points PP and QQ with parameters t=1t=1 and t=−12t=-\frac12 respectively. Find the area bounded by CC and ll, correct to 3 significant figures.

Explore this idea

Hint

  1. Find the coordinates of PP and QQ.

  2. Is yy steadily increasing along the arc from QQ to PP? If so, use horizontal strips and integrate x dyx\,dy.

  3. The region is a trapezium (between ll and the yy-axis) with the part between CC and the yy-axis removed.

Answer · A

A=1701/320≈5.32A=1701/320\approx5.32 square units (3 s.f.).

Source-selected arc −1/2≤t≤1. Full real-parameter curve also meets the line at t=7, giving a different additional region.

Explore the idea

Horizontal strips between the curve and the line

The horizontal axis here is y, and the vertical axis is x. Each strip is a thin horizontal slice of the region, with width (line − curve).

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 y* = -1.625.

zero heightupper = lowerwidth Δy
Zoomed-in rectangle: width and height are scaled independently. The sample sets its upper and lower values across the whole strip.
Sample y*
-1.625
Width Δy
1.4063
Upper value
0.75
Lower value
0.75
Height: upper − lower
0
Strip area
0

1.4063 ×(0.75 − 0.75)≈ 0

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

Strip width
1.406
5.625 ÷ 4
Sum of strips
5.01757
Exact, 1701/320
5.31562
Error
−0.29806
5.61% too small

How the sum improves as n grows

  • C, drawn with y along the bottom and x up the side
  • L, the straight line through Q and P, written the same way
  • Each has width (x on l − x on C)
  • Exact area 1701/320 ≈ 5.3156
Key idea

Seen with yy along the axis, the curve is a single-valued function and the region is a simple area between two graphs. The strips settle on 1701320≈5.32\frac{1701}{320}\approx5.32.

Ideas to try
  1. Imagine turning your head sideways. The picture is the region in the question.
  2. Why does the curve touch the horizontal axis (x = 0) at y = 0?
  3. Compare the left-edge and right-edge rules at n = 6.
Takeaway

Pick the variable that does not turn back

Before integrating a parametric curve, check which of xx and yy is monotonic along the arc. Integrating with respect to that one means the curve is a function, and the area formula ∫(right−left) dy\int(\text{right}-\text{left})\,dy or ∫(top−bottom) dx\int(\text{top}-\text{bottom})\,dx applies directly.