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

Choose a part to attempt. Each page keeps the full question available and loads only that part’s working and exploration.

  1. (i) Curve sketching →
  2. (ii) Area between curves →
  3. (iii) Volume of revolution →
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.