The solid lies above the region D in the x y plane bounded by the circle x 2 y 2 = r 2, so the volume is given by the integral ∫ ∫ D f ( x, y) d A = ∫ − r r ∫ − r 2 − y 2 r 2 − y 2 f ( x, y) d x d y Therefore the required volume of the solid is ∫ − r r ∫ − r 2Oliver Knill, Harvard Summer School, 10 Chapter 2 Surfaces and Curves Section 21 Functions, level surfaces, quadrics A function of two variables f(x,y) is2 We can describe a point, P, in three different ways Cartesian Cylindrical Spherical Cylindrical Coordinates x = r cosθ r = √x2 y2 y = r sinθ tan θ = y/x z = z z = z Spherical Coordinates
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