Subtract a triangle from an ellipse quadrant

Ellipse and line bounding R; labels −a, a, b, −b, O, x, ySource-given figure redrawn with D3. Ellipse and line bounding R; labels −a, a, b, −b, O, x, y. Only labels supplied in the original question are included.xyOR−aab−by/b − x/a = 1x²/a² + y²/b² = 1
Question diagram 1
Ellipse and line bounding R; labels −a, a, b, −b, O, x, ySource-given figure redrawn with D3. Ellipse and line bounding R; labels −a, a, b, −b, O, x, y. Only labels supplied in the original question are included.xyOR−aab−by/b − x/a = 1x²/a² + y²/b² = 1

Given ellipse and line enclosing R, labelled semi-axes a and b.

(i) R is bounded by the ellipse x2/a2+y2/b2=1x^2/a^2+y^2/b^2=1 and the line y/b−x/a=1y/b-x/a=1, as shown. Show that its area is

−12ab+∫−a0baa2−x2dx.-\frac12ab+\int_{-a}^0\frac ba\sqrt{a^2-x^2}dx.

Hence, substituting x=asin⁡θx=a\sin\theta, find the exact area in terms of a,ba,b. marks

(ii) Find the volume when R is rotated completely about the yy-axis, in the form ka2bka^2b with constant kk. marks

[2010/MJC/Prelim/II/5]

Total marks: 12

Solution

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

  1. (i)·1 Setting up an area →
  2. (i)·2 Trigonometric substitution →
  3. (ii) Volume of revolution →
Takeaway

Check with known solids

Half an ellipsoid has volume 23πa2b\frac23\pi a^2b and a cone of radius aa, height bb has 13πa2b\frac13\pi a^2b. Quoting both from memory gives the answer to (ii) in one line, and it is a good check on the integral.