Find the region before you choose the radii

Curve C: y=3−10/(x2−2x+4)y=3-10/(x^2-2x+4).

(i) Without a calculator, determine its exact stationary point. marks

(ii) Sketch C, stating clearly any asymptotes. marks

R is bounded by C, 3y+16x=153y+16x=15 and the yy-axis.

(iii) Find its exact area. marks

(iv) Find the volume when R is rotated 2π2\pi radians about the yy-axis. marks

[2021/EJC/Prelim/II/4]

Total marks: 10

Solution

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

  1. (i) Stationary points →
  2. (ii) Curve sketching →
  3. (iii) Area between curves →
  4. (iv) Volume of revolution →
Takeaway

Sketch first, then choose radii

The region touches the axis only for y≥12y\ge\frac12. Below that the curve moves away from the axis, which is what creates the hollow. Seeing that on a sketch is what tells you whether you need one integral or a subtraction.