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DOUBLE INTEGRAL HELP!!?

Convert the integral below to polar coordinates and evaluate the integral.∫(0 to 5/√2) ∫(y to √(25−y^2)) xy dxdy
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Oswald Hills

Feb 20, 2025

The region of integration is x = y to x = √(25−y^2) with y in [0, 5/√2].==> Inside x^2 + y^2 = 25 in the first quadrant under y = x.==> Inside r = 5 with θ in [0, π/4] using polar coordinates.So, ∫(y = 0 to 5/√2) ∫(x = y to √(25 – y^2)) xy dx dy= ∫(θ = 0 to π/4) ∫(r = 0 to 5) (r cos θ)(r sin θ) * (r dr dθ), via polar coordinates= ∫(θ = 0 to π/4) sin θ cos θ dθ * ∫(r = 0 to 5) r^3 dr = (1/2) sin^2(θ) {for θ = 0 to π/4} * (1/4)r^4 {for r = 0 to 5}= 625/16.I hope this helps!... Show More
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