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 ρ2sinφ\rho^2 \sin \varphi
    • An Example
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Examples
Lessons
  1. Convert the following integral into spherical coordinates

    323294x294x23x2+3y29x2y2x3+xy2+xz2dzdydx\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

      202x20x2+y24x2y2x+y+dzdydx\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+2y2dV\int\int\int_{E}2x^{2} + 2y^{2} \, dV \, where EE is the region portion of x2+y2+z2=1x^{2} + y^{2} +z^{2} = 1\, with z0z \geq 0.
        1. Evaluating Triple Integrals in Spherical Coordinates
          Evaluate E3dV\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=ρsinφr = \rho \sin \varphi
          z=ρcosφz = \rho \cos \varphi
          z2+r2=ρ2z^2 + r^2 = \rho^2

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

          x=ρsinφcosθx = \rho \sin \varphi \cos \theta
          y=ρsinφsinθy = \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:

          aρb a \leq \rho \leq b
          αθβ\alpha \leq \theta \leq \beta
          δφγ\delta \leq \varphi \leq \gamma

          Then the conversion would be:

          Ef(x,y,z)dV=δγαβabf(ρsinφcosθ,ρsinφsinθ,ρcosφ)ρ2sinφdpdθdφ\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