Using integration, find the area enclosed by the ellipse 16x² + 25y² = 400.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Ellipse 16x² + 25y² = 400. Dividing by 400 gives x²/25 + y²/16 = 1, so a = 5 and b = 4. To find: the area it encloses.
2 · Draw the figure. Draw the oval through (±5, 0) and (0, ±4). Shade the first-quadrant part and draw a vertical strip.
3 · Write the formula, rule or theorem. Upper half y = (4/5)√(25 − x²). By symmetry, A = 4 ∫ y dx from 0 to 5.
4 · Integrate.
A = (16/5) ∫ √(25 − x²) dx from 0 to 5
= (16/5)[(x/2)√(25 − x²) + (25/2) sin⁻¹(x/5)] from 0 to 5
5 · Put in the limits. = (16/5)[(0 + (25/2)(π/2)) − 0] = (16/5)(25π/4)
6 · Simplify. A = 20π
7 · Check and box the answer. Check: πab = π × 5 × 4 = 20π. Area = 20π sq units.
Scoring tip: Write the standard form and the values of a and b first. Show the symmetry factor 4.
