Using integration, find the area of the region bounded by the parabola y² = 8x and its latus rectum.
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. Parabola y² = 8x, so 4a = 8 and a = 2. The latus rectum is x = 2. To find: the area between them.
2 · Draw the figure. Draw y² = 8x opening to the right from O. Draw x = 2, meeting the curve at (2, 4) and (2, −4). Shade the region and draw a vertical strip.
3 · Write the formula, rule or theorem. By symmetry about the x-axis, A = 2 ∫ y dx from 0 to 2, with y = √(8x) = 2√2 × √x.
4 · Integrate.
A = 2 × 2√2 × (2/3)[x√x] from 0 to 2
= (8√2/3) × 2√2
= 32/3
5 · Check and box the answer. Check with 8a²/3 = 8 × 4/3 = 32/3. Area = 32/3 sq units.
Scoring tip: Find a from 4a first. The latus rectum is x = a, not x = 4a.
