Find the (surface) area cut out of the cylinder x^2+z^2=9 by the cylinder x^2+y^

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example S is the upper cap cut from the sphere x^2+y^2+z^2=25 by the cylinder x^2+y^2=9? find the surface area of the surface s ans Since z > 0, we see that z = v(25 - x^2 - y^2). The region of integration is the interior of x^2 + y^2 = 9. Using Cartesian Coordinates, the surface area equals ?? v[1 + (z_x)^2 + (z_y)^2] dA = ?? v[1 + (-x/v(25 - x^2 - y^2))^2 + (-y/v(25 - x^2 - y^2))^2] dA = ?? v[1 + (x^2 + y^2)/(25 - x^2 - y^2)] dA = ?? v[25/(25 - x^2 - y^2)] dA = ?? 5 dA /v(25 - x^2 - y^2) Now, convert to polar coordinates: ?(r = 0 to 3) ?(? = 0 to 2p) 5 * r d? dr/v(25 - r^2) = ?(r = 0 to 3) 5 * 2pr dr/v(25 - r^2) = -5pv(25 - r^2) {for r = 0 to 3} = 5p.

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