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Triple integrals in cylindrical coordinates

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Intros
Lessons
  1. Triple Integrals in Cylindrical Coordinates Overview:
  2. Triple Integrals in Cylindrical Coordinates
    • Polar Coordinates →\to Cylindrical Coordinates
    • All xx's & yy's change to rr's & θ\theta
    • zz stays the same
  3. An Example of Converting to Cylindrical Coordinates
    • All xx's & yy's change to rr's & θ\theta
    • The variable zz stays the same
    • Add an extra rr
    • Integrate
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Examples
Lessons
  1. Converting to Cylindrical Coordinates
    Convert the following triple integral to cylindrical coordinates

    ∫−20∫04−x2∫2x2+2y2−4x+y9−x2−y2 dz dy dx\large \int_{-2}^{0}\int_{0}^{\sqrt{4 - x^{2}}} \int_{2x^{2} + 2y^{2}-4}^{x+ y} \sqrt{9 - x^{2} - y^{2}}\, dz \,dy\, dx
    1. Convert the following triple integral to cylindrical coordinates

      ∫−33∫−9−y29−y2∫3z−52 zIn(x2+y2) dx dz dy\large \int_{-3}^{3}\int_{-\sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} \int_{3z- 5}^{2} \, zIn(x^{2} + y^{2})\, dx \,dz\, dy
      1. Converting & Integrating
        Evaluate  ∫∫∫E2dV  \, \int\int\int_{E} 2dV \, where  E  \, E \, is the region bounded by  z=x2+y2−2 \, z = x^{2} + y^{2} - 2 \, and  z=6−x2−y2 \, z = 6 - x^{2} - y^{2} .
        1. Evaluate  ∫∫∫E1dV  \, \int\int\int_{E} 1dV \, where  E  \, E \, is the region bounded by  z=4,z=x−y−3 \, z = 4, z = x - y - 3 \, and inside  x2+y2=4 \, x^{2} + y^{2} = 4 .
          Topic Notes
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          Notes:

          Recall that when converting from Cartesian Coordinates to Polar Coordinates with double integrals we do the following:

          ∫∫Df(x,y)dA=∫θ=αθ=β∫r=g1(θ)r=g2(θ)f(rcos⁡θ,rsin⁡θ)rdrdθ\int \int_D f(x,y)dA = \int^{\theta=\beta}_{\theta=\alpha} \int^{r=g_2(\theta)}_{r=g_1(\theta)} f(r \cos \theta, r \sin \theta ) r dr d\theta

          With triple integrals, it is very similar. Instead of calling it polar coordinates, we call it cylindrical coordinates.

          Suppose we want to triple integrate f(x,y,z)f(x,y,z) in cylindrical coordinates in the following region of EE.

          α≤θ≤β \alpha \leq \theta \leq \beta
          g1(θ)≤r≤g2(θ) g_1 (\theta) \leq r \leq g_2 (\theta)
          h1(rcos⁡θ,rsin⁡θ)≤z≤h2(rcos⁡θ,rsin⁡θ)h_1 (r \cos \theta, r \sin \theta) \leq z \leq h_2(r \cos \theta, r \sin \theta)

          Then

          ∫∫∫Ef(x,y,z)dV=∫αβ∫g1(θ)g2(θ)∫h1(rcos⁡θ,rsin⁡θ)h2(rcos⁡θ,rsin⁡θ)f(rcos⁡θ,rsin⁡θ,z)rdzdrdθ \int \int \int_E f(x,y,z) dV = \int^{\beta}_{\alpha} \int^{g_2(\theta )}_{g_1(\theta )} \int^{h_2 (r \cos \theta, r \sin \theta)}_{h_1 (r \cos \theta, r \sin \theta)} f(r \cos \theta, r \sin \theta, z) rdzdrd\theta