parametrize the surface cut from the hemisphere x^2 + y^2 + z^2 =4 , by the cylinder x^2 + y^2 = 2x.
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Assuming that we're discussing the upper hemisphere so that z = √(4 - x^2 - y^2)
Use the polar parameterization: R(φ, θ) = <2 cos θ , 2 sin θ, √(4 - r^2)>.
As for for the region, writing x^2 + y^2 = 2x in polar coordinates yields
r^2 = 2r sin θ ==> r = 2 sin θ, which is completely traced out for θ in [0, π].
Hence, an answer is R(φ, θ) = <2 cos θ , 2 sin θ, √(4 - r^2)>, where
r = 0 to r = 2 sin θ with θ in [0, π].
I hope this helps!
Use the polar parameterization: R(φ, θ) = <2 cos θ , 2 sin θ, √(4 - r^2)>.
As for for the region, writing x^2 + y^2 = 2x in polar coordinates yields
r^2 = 2r sin θ ==> r = 2 sin θ, which is completely traced out for θ in [0, π].
Hence, an answer is R(φ, θ) = <2 cos θ , 2 sin θ, √(4 - r^2)>, where
r = 0 to r = 2 sin θ with θ in [0, π].
I hope this helps!
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Minor correction: R(r, θ)
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