Rotate the region between a bell curve and a line

(i) Sketch y=2/[1+(x−2)2]y=2/[1+(x-2)^2] and y=1y=1 on the same diagram, showing clearly the coordinates of the stationary point and intersection points. marks

(ii) Find the exact area of region R enclosed by the curves and x=1,x=3x=1,x=3. marks

(iii) Find the volume generated when R is rotated 360∘360^\circ about the yy-axis. marks

[2011/MI/Prelim/II/2]

Total marks: 9

Solution

If this part uses an earlier result, revisit (i) Curve sketching or open all working together.

(ii)

Area between curves

Find the exact area of the region RR enclosed by the two curves and the lines x=1x=1 and x=3x=3.

Explore this idea

Hint

  1. Each thin vertical strip of RR has height (curve) − (line).

  2. Integrate that height from x=1x=1 to x=3x=3.

  3. “Exact” means the answer stays in terms of π\pi, so look for an arctangent.

Answer · (ii)

Area =π−2=\pi-2.

Explore the idea

Open interactive exploration

An integral is the limit of a sum of strips

Approximate R with n vertical strips. Each has width 2/n and height curve − line. Add more strips and watch the total settle on the exact area.

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* = 1.

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*
1
Width Δx
0.5
Upper value
1
Lower value
1
Height: upper − lower
0
Strip area
0

0.5 ×(1 − 1)≈ 0

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

Strip width
0.500
2.000 ÷ 4
Sum of strips
1.10000
Exact, π − 2
1.14159
Error
−0.04159
3.64% too small

How the sum improves as n grows

  • Rectangles of width 2/n. Each height is curve − line, measured at the left edge, midpoint or right edge
  • Exact area π − 2 ≈ 1.14159
  • Strip total for every n from 1 to 40. The dot marks your current n
Key idea

Strip totals converge to π−2≈1.14159\pi-2\approx1.14159, and the integral is exactly that limit. Midpoint strips get there fastest because each strip's overestimate on one side nearly cancels its underestimate on the other.

Ideas to try
  1. Set n = 2 with the left-edge rule. Why do the strips miss so much of R?
  2. Switch to midpoints and compare the error at n = 10 with the left-edge rule.
  3. Drag n up to 40 and watch the error shrink towards zero.
Takeaway

Connect the parts: area × distance travelled

By the symmetry from (i), the centre of RR is at x=2x=2. Each turn carries that centre around a circle of length 2π×22\pi\times2. Multiply by the area from (ii) and you get the volume from (iii):

V=2π⋅2⋅(π−2)=4π(π−2)V=2\pi\cdot2\cdot(\pi-2)=4\pi(\pi-2)

This is Pappus’s centroid theorem. It is not on the syllabus, but it is a quick independent check of your answer.