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7329733 - We are asked to find the volume of the solid...

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We are asked to find the volume of the solid bounded by the sphere = ρ 6cosφ and the elliptical cone = + z 2x2 3y2 In spherical coordinates, = , = , = , = + + x ρsinφcosθ y ρsinφsinθ z ρcosφ and ρ x2 y2 z2 . Change the equation of the sphere to Cartesian coordinates: + + = + + x2 y2 z2 6zx2 y2 z2 + + = x2 y2 z2 6z + + - + = x2 y2 z2 6z 9 9 + + - = x2 y2 z 32 9 which is the equation of a sphere with radius 3 and center (0,0,3). It's projection onto the xy-plane is a circle of radius 3 and center at the origin . As for the elliptical cone, it's projection onto the xy plane is an ellipse with center at the origin. So, the projection on the xy-plane bounded by both this circle and ellipse is going to be region that goes completely around the xy plane. Thus, the bounds of θ in spherical coordinates of the 3D solid will be 0 to 2π . The lower bound of φ is 0. To find the upper bound, switch the equation of the elliptical cone to spherical coordinates and simplify: = + ρcosφ 2ρ2sin2φcos2θ 3ρ2sin2φsin2θ
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