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Surface of a Cylinder inside a Sphere
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[QUOTE="mishima, post: 6053159, member: 271564"] [h2]Homework Statement [/h2] This is a problem from Boas, Mathematical Methods of the Physical Sciences chapter 5, section 5, number 6. Find the area of the cylinder x^2+y^2-y=0 inside the sphere x^2+y^2+z^2=1. [h2]Homework Equations[/h2] This section deals with projecting curved areas onto a coordinate plane and evaluating using double integrals. The projected surface is related to the curved surface by dxdy=dA cos γ or dA=sec γ dxdy where γ is measured off the 3rd axis (the one not being projected onto). Also, sec γ = grad φ / (dφ/dz) where φ is the equation for the curved surface. So, the differential area can be evaluated through a double integral after choosing the right bounds. [h2]The Attempt at a Solution[/h2] In this problem, the cylinder axis is parallel to the z axis, so a projection onto the xy plane will not work. Instead, we can cycle through the axes and modify the equations like: dA=sec γ dydz sec γ = grad φ / (dφ/dx) Where the latter becomes, for our cylinder, sec γ = 1/(y-y^2)^(1/2) The equation for the sphere gives the bounds for the double integral, y going from -(1-z^2)^(1/2) and +(1-z^2)^(1/2) while z is just from 0 to 1. I can evaluate one, but not both of the integrals with what I know. I have also tried changing coordinate systems to polar but there was not a simplification. The answer is supposedly 4, so I feel like I'm missing something obvious. [/QUOTE]
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Surface of a Cylinder inside a Sphere
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