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Area bounded by x²y=a²(a−y) and x-axis
Short answer
Rewrite as y = a³/(x²+a²). The area is ∫_{-∞}^∞ a³/(x²+a²) dx = 2a³ ∫₀^∞ dx/(x²+a²) = πa².
Find the area bounded by the curve x²y = a²(a−y) and the X-axis.
Curve: x²y = a²(a − y). Solve for y:
x²y + a²y = a³, so y = a³/(x² + a²) (the Witch of Agnesi).
The curve lies above the x-axis (y > 0), is symmetric about the y-axis, and y → 0 as x → ±∞; so the x-axis is an asymptote. The area between the curve and the x-axis is
A = ∫_(−∞)^(∞) a³/(x² + a²) dx
= 2a³ ∫₀^∞ dx/(x² + a²)
= 2a³ · (1/a)[tan⁻¹(x/a)]₀^∞
= 2a² · (π/2)
Area = πa² square units.
✓ Answer checked by Claude Opus 5.5 (high effort)
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