Using integration, find the area of the region of the circle x² + y² = 16 that lies between the lines x = −2 and x = 2.
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. Circle x² + y² = 16 (centre O, radius 4) and the lines x = −2 and x = 2. To find: the area of the circle between these lines.
2 · Draw the figure. Draw the circle through (±4, 0) and (0, ±4). Draw x = −2 and x = 2; they meet the circle at (±2, ±2√3). Shade the band and draw a vertical strip.
3 · Write the formula, rule or theorem.
By symmetry about both axes, A = 4 ∫ √(16 − x²) dx from 0 to 2.
∫ √(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a), with a = 4.
4 · Integrate. A = 4[(x/2)√(16 − x²) + 8 sin⁻¹(x/4)] from 0 to 2
5 · Put in the limits.
At x = 2: 1 × √12 + 8 × π/6 = 2√3 + 4π/3
At x = 0: 0
6 · Simplify. A = 4(2√3 + 4π/3) = 8√3 + 16π/3
7 · Check and box the answer. 8√3 + 16π/3 ≈ 30.6, less than the whole circle, 16π ≈ 50.3. Area = (8√3 + 16π/3) sq units.
Scoring tip: Say in words why you multiply by 4, and write the standard integral before you use it.
