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Triple integrals in spherical coordinates

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Intros
Lessons
  1. Triple Integrals in Spherical Coordinates Overview:
  2. Spherical Coordinates
    • (x,y,z)→(ρ,θ,φ)(x,y,z) \to (\rho, \theta, \varphi )
    • Graph of the coordinates in 3D
  3. Equations to Convert from Cartesian to Spherical
    • Pythagoras
    • Trig Ratios
  4. Triple Integrals in Spherical Coordinates
    • Region EE
    • Add an extra ρ2sin⁡φ\rho^2 \sin \varphi
    • An Example
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Examples
Lessons
  1. Convert the following integral into spherical coordinates

    ∫−3232∫−94−x294−x2∫3x2+3y29−x2−y2 x3+xy2+xz2 dz dy dx\large \int_{-\frac{3}{2}}^{\frac{3}{2}}\int_{-\sqrt{\frac{9}{4}} - x^{2}}^{\sqrt{\frac{9}{4} - x^{2}}}\int_{\sqrt{3x^{2} + 3y^{2}}}^{\sqrt{9 - x^{2} - y^{2}}} \, x^{3} + xy^{2} + xz^{2} \,dz \,dy \,dx
    1. Convert the following integral into spherical coordinates

      ∫−20∫−2−x20∫x2+y24−x2−y2 x+y+dz dy dx\large \int_{ -\sqrt{2} }^{0} \int_{-\sqrt{2 - x^{2}}}^ {0} \int_{\sqrt{x^{2} + y^{2}}}^ {\sqrt{4 - x^{2} - y^{2}}} \, x + y + dz \, dy \,dx
      1. Evaluating Triple Integrals in Spherical Coordinates
        Evaluate ∫∫∫E2x2+2y2 dV \int\int\int_{E}2x^{2} + 2y^{2} \, dV \, where EE is the region portion of x2+y2+z2=1 x^{2} + y^{2} +z^{2} = 1\, with z≥0z \geq 0.
        1. Evaluating Triple Integrals in Spherical Coordinates
          Evaluate ∫∫∫E3 dV \int\int\int_{E}3 \, dV \, where EE is the region bounded by x2+y2+z2=4x^{2} + y^{2} +z^{2} = 4 and z=−x2+y2z = -\sqrt{x^{2} + y^{2}} .
          Topic Notes
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          Notes:

          Spherical Coordinates

          There are times where instead of Cartesian coordinates, we use spherical coordinates for triple integrals. For spherical coordinates, instead of x,y,zx,y,z's, we have ρ,θ,φ\rho, \theta, \varphi's. In other words,

          (x,y,z)→(ρ,θ,φ)(x,y,z) \to (\rho, \theta, \varphi )

          spherical coordinates

          Equations to Convert From Cartesian to Spherical

          From the graph, we can obtain the following equations which will be useful for converting spherical to cartesian, or vice versa:

          r=ρsin⁡φr = \rho \sin \varphi
          z=ρcos⁡φz = \rho \cos \varphi
          z2+r2=ρ2z^2 + r^2 = \rho^2

          Using the equations from past sections, we can also obtain more equations:

          x=ρsin⁡φcos⁡θx = \rho \sin \varphi \cos \theta
          y=ρsin⁡φsin⁡θy = \rho \sin \varphi \sin \theta
          ρ2=x2+y2+z2\rho^2 = x^2 + y^2 + z^2


          Triple Integrals in Spherical Coordinates

          Suppose we want to convert a triple integral of f(x,y,z)f(x,y,z) in Cartesian Coordinates to spherical Coordinates on region EE. Let region EE be:

          a≤ρ≤b a \leq \rho \leq b
          α≤θ≤β\alpha \leq \theta \leq \beta
          δ≤φ≤γ\delta \leq \varphi \leq \gamma

          Then the conversion would be:

          ∫∫∫Ef(x,y,z)dV=∫δγ∫αβ∫abf(ρsin⁡φcos⁡θ,ρsin⁡φsin⁡θ,ρcos⁡φ)ρ2sin⁡φdpdθdφ\int \int \int_E f(x,y,z)dV = \int_\delta^\gamma \int_\alpha^\beta \int_a^b f(\rho \sin \varphi \cos \theta, \rho \sin \varphi \sin \theta, \rho \cos \varphi)\rho^2 \sin \varphi d p d\theta d\varphi