A-Level Definite integrals: worked solution

Question

Curve C comes from rotating ellipse S about the origin, with x = cos t - 2 sin t, y = cos t + 2 sin t. Give a possible cartesian equation of S and hence the total area of the flower bed.

Start from the parametric form of C: x = cos t − 2 sin t y = cos t + 2 sin t Adding and subtracting removes one trigonometric function at a time: x + y = 2 cos t, so cos t = (x + y)/2 y − x = 4 sin t, so sin t = (y − x)/4 Using cos²t + sin²t = 1 gives C's cartesian equation: ((x + y)/2)² + ((y − x)/4)² = 1. However S is the ellipse before the rotation. Group the parametric form by its two constant vectors: r = (cos t)(1, 1) + (sin t)(−2, 2) The vectors (1, 1) and (−2, 2) are perpendicular, since (1)(−2) + (1)(2) = 0, and their lengths are |(1, 1)| = √2 and |(−2, 2)| = 2√2. So C is an ellipse with semi-axes √2 and 2√2, tilted from the coordinate axes. A rotation about O does not change lengths, so S has the same semi-axes but lying along the axes: x²/(2√2)² + y²/(√2)² = 1 x²/8 + y²/2 = 1. For the area, make y the subject in the first quadrant: y² = 2(1 − x²/8) = 2 − x²/4 y = √(2 − x²/4), for 0 ≤ x ≤ √8 The ellipse is symmetric in both axes, so the total area is four times the first-quadrant part: Area = 4 ∫₀^√8 √(2 − x²/4) dx That integral is a quarter of the ellipse, so it equals (1/4)πab = (1/4)π(2√2)(√2) = π, giving Area = 4π = 12.566... Area ≈ 12.6 m² (3 s.f.).