Triple integrals in cylindrical coordinates

Definition

Triple integrals in cylindrical coordinates are a way to calculate the volume under a surface or the accumulation of a quantity over a three-dimensional region using a coordinate system that simplifies the representation of objects with circular symmetry. This system uses three variables: radial distance (r), angle (θ), and height (z), making it easier to work with problems involving cylinders or circular shapes. By converting from Cartesian to cylindrical coordinates, the integration process can become more straightforward for certain geometric scenarios.

5 Must Know Facts For Your Next Test

1. When using triple integrals in cylindrical coordinates, the volume element dV is expressed as r \, dr \, d heta \, dz, where r is the radial distance from the origin.
2. The limits of integration for r typically range from 0 to the radius of the cylinder, while θ ranges from 0 to 2π, and z spans the height of the cylinder.
3. The transformation from Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z) is done using x = r \cos(θ), y = r \sin(θ), and z = z.
4. Triple integrals can be used to find volumes of irregular shapes by defining appropriate limits of integration based on the geometry of the region.
5. Cylindrical coordinates are particularly useful when dealing with problems involving rotational symmetry, making calculations more manageable.

Review Questions

• How do you set up a triple integral in cylindrical coordinates for a cylinder with a specified height and radius?
  • To set up a triple integral for a cylinder, first identify the limits of integration for each variable. For r, the limits will typically range from 0 to the radius of the cylinder. For θ, it will range from 0 to 2π since we want to encompass the full circular cross-section. Finally, for z, you'll set limits based on the height of the cylinder. The integral can then be expressed as ∫∫∫ r \, dz \, dr \, d heta with these limits.
• Explain how you would convert a triple integral from Cartesian coordinates to cylindrical coordinates.
  • To convert a triple integral from Cartesian to cylindrical coordinates, you replace x and y with their cylindrical equivalents: x = r \cos(θ) and y = r \sin(θ). The volume element also changes; in cylindrical coordinates, dV = r \, dr \, d heta \, dz. Make sure to adjust the limits of integration based on the geometry of your specific problem. The resulting integral will reflect these changes.
• Evaluate the triple integral of f(x, y, z) = xyz over the volume of a cylinder with radius 3 and height 5 using cylindrical coordinates.
  • To evaluate this integral, first convert f(x, y, z) into cylindrical coordinates: f(r, θ, z) = (r \cos(θ))(r \sin(θ))(z) = r^2 \cos(θ) \sin(θ) z. Next, set up your triple integral: ∫ from z=0 to z=5 ∫ from θ=0 to θ=2π ∫ from r=0 to r=3 (r^2 \cos(θ) \sin(θ) z) \, r \, dr \, d heta \, dz. After calculating this integral step-by-step by first integrating with respect to r, then θ, and finally z, you’ll find the total accumulation over the cylinder’s volume.

Related terms

Cylindrical Coordinates: A three-dimensional coordinate system that extends polar coordinates by adding a height dimension, using (r, θ, z) to represent points in space.

Jacobian: A determinant used in changing variables during integration, essential for converting volume elements when switching from Cartesian to cylindrical coordinates.

Volume Element: The infinitesimal volume element in cylindrical coordinates, represented as dV = r \, dr \, d heta \, dz, which incorporates the radial distance and angle.