Triple integrals in cylindrical coordinates

Triple integrals in cylindrical coordinates

Lessons

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

  • Introduction
    Triple Integrals in Cylindrical Coordinates Overview:
    a)
    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

    b)
    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

  • 1.
    Converting to Cylindrical Coordinates
    Convert the following triple integral to cylindrical coordinates

    ∫−20∫04−x2∫2x2+2y2−4x+y9−x2−y2dzdydx\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
  • 2.
    Convert the following triple integral to cylindrical coordinates

    ∫−33∫−9−y29−y2∫3z−52zIn(x2+y2)dxdzdy\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
  • 3.
    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} .
  • 4.
    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 .
Do better in math today
Get Started Now
5.
Multiple Integrals
5.1
Double integrals over a rectangular region
5.2
Double integrals over a general region
5.3
Double integrals in polar coordinates
5.4
Triple integrals
5.5
Triple integrals in cylindrical coordinates
5.6
Triple integrals in spherical coordinates
Calculus 3