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

(i)

Curve sketching

Sketch y=21+(x−2)2y=\dfrac{2}{1+(x-2)^2} and y=1y=1 on the same diagram. Label the stationary point and the intersection points.

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Hint

  1. Where is yy largest? Look at when the denominator 1+(x−2)21+(x-2)^2 is smallest.

  2. Where does the curve meet y=1y=1? Solve 21+(x−2)2=1\frac{2}{1+(x-2)^2}=1.

  3. What happens as x→±∞x\to\pm\infty, and where does the curve cross the yy-axis?

Bell curve area and washer geometry
Bell curve area and washer geometryScroll across to see the full diagram.

Answer · (i)

Maximum (2,2)(2,2); intersections (1,1),(3,1)(1,1),(3,1); asymptote y=0y=0.

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Open interactive exploration

Add each feature an examiner looks for

Start with the bare curve and line. Switch on each feature to see where it goes and how you would find it.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Bell curve y = 2/[1+(x−2)²]
  • Horizontal line y = 1
  • Line of symmetry x = 2
Ideas to try
  1. Switch on “Meets y = 1” first. Could you predict both x-values from the symmetry alone?
  2. Hover or tap along the curve to read off coordinates.
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.