Application of Integration: Plane Area

Question

Show your mathematical maturity by giving COMPLETE complete solution to each of the

following items.

1. Find the area bounded by the curve y² = 4x and the line 2x + y = 4.

2. Find the area bounded by the curve 5y² = 16x and the curve y² = 8x − 24.

Answer 1

1.y² = 4x and the line 2x + y = 4; x = y²/4; y²/2 + y = 4; y² + 2y - 8 = 0; y = - 4 or y = 2.

Area A = ∫_-4²(2 - y/2 - y²/4)dy = (2y - y²/4 - y³/12)_-4² = (4 - 4/4 - 8/12) - (- 8 - 4 + 64/12) = 3 - 2/3 + 12 - 16/3 = 9

2.5y² = 16x and the curve y² = 8x − 24. x = 5y²/16; y² = 5y²/2 - 24; 3y² = 48; y = ± 4

Area A = ∫_-4⁴(y²/8 + 3 - 5y²/16)dy = 2∫₀⁴(3 - 3y²/16)dy = 2(3y - y³/16)₀⁴ = 2(12 - 4) = 16