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13.7 Notes - Calculus III Section 13.7 – Triple Integrals...

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Unformatted text preview: Calculus III Section 13.7 – Triple Integrals in Cylindrical and Spherical Coordinates As we have already seen many integrals are easier to integrate if we use cylindrical or spherical coordinates instead of rectangular coordinates. In this section we will just practice integrating in the other coordinate systems. Cylindrical Coordinates ∫ Q ∫ f∫( x, y, z )dV = θ 2 g 2 ( θ ) h2 ( r cos θ , r sin θ ) ∫∫ ∫ f (r cosθ , r sin θ , z )rdzdrdθ θ1 g1 ( θ ) h1 ( r cos θ , r sin θ ) Since we have already integrated many iterated integrals in cylindrical coordinates we will just go ahead and practice one in spherical coordinates together. The cylindrical coordinate integrals in this section are no different than the ones we have already done. Spherical Coordinates ∫ Q ∫ f∫( x, y, z )dV = Example: θ 2 φ2 ρ 2s ∫ ∫ ∫ f ( ρ sin φ cosθ , ρ sin φ sin θ , ρ cos φ ) ρ θφ ρ 1 π /4 π /4 cos θ 0 0 ∫∫∫ 0 1 1 ρ 2 sin φ cos φ dρ dθ dφ 2 sin φ dρ dφ dθ ...
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