Math Problem Statement

Convert \int_{0}^{1} \int_{0}^{\pi/4} \int_{0}^{2\pi} {\rho^{3}\sin{(2\phi)} ,d\theta ,d\phi ,d\rho} to cartesian coordinates

Solution

To convert the given triple integral from cylindrical coordinates to Cartesian coordinates, we need to follow these steps:

Step 1: Recognize the coordinate system.

The original triple integral is expressed in cylindrical coordinates. Cylindrical coordinates (ρ,ϕ,θ)(\rho, \phi, \theta) are related to Cartesian coordinates (x,y,z)(x, y, z) through the following relations:

  • x=ρcos(θ)x = \rho \cos(\theta)
  • y=ρsin(θ)y = \rho \sin(\theta)
  • z=ρsin(ϕ)z = \rho \sin(\phi)
  • ρ=x2+y2\rho = \sqrt{x^2 + y^2}
  • tan(θ)=yx\tan(\theta) = \frac{y}{x}

Here, ρ\rho represents the radial distance in the xy-plane, ϕ\phi is the angle between the radius and the z-axis (the "polar angle"), and θ\theta is the azimuthal angle (rotation in the xy-plane).

Step 2: Rewriting the integrand.

The given integral is: 010π/402πρ3sin(2ϕ)dθdϕdρ\int_{0}^{1} \int_{0}^{\pi/4} \int_{0}^{2\pi} \rho^{3} \sin(2\phi) \,d\theta \,d\phi \,d\rho We want to express the integrand ρ3sin(2ϕ)\rho^3 \sin(2\phi) and the limits of integration in Cartesian coordinates.

Step 3: Convert the integrand.

In Cartesian coordinates:

  • ρ=x2+y2\rho = \sqrt{x^2 + y^2}, so ρ3\rho^3 becomes (x2+y2)3/2(x^2 + y^2)^{3/2}.
  • The function sin(2ϕ)\sin(2\phi) depends on ϕ\phi, which is the angle between ρ\rho and the z-axis. In Cartesian coordinates, we can express ϕ\phi through the relation: sin(ϕ)=zx2+y2+z2\sin(\phi) = \frac{z}{\sqrt{x^2 + y^2 + z^2}} To express sin(2ϕ)\sin(2\phi), we use the double-angle identity: sin(2ϕ)=2sin(ϕ)cos(ϕ)\sin(2\phi) = 2\sin(\phi)\cos(\phi) where: cos(ϕ)=x2+y2x2+y2+z2andsin(ϕ)=zx2+y2+z2\cos(\phi) = \frac{\sqrt{x^2 + y^2}}{\sqrt{x^2 + y^2 + z^2}} \quad \text{and} \quad \sin(\phi) = \frac{z}{\sqrt{x^2 + y^2 + z^2}} Hence, we get: sin(2ϕ)=2(zx2+y2+z2)(x2+y2x2+y2+z2)\sin(2\phi) = 2 \left(\frac{z}{\sqrt{x^2 + y^2 + z^2}}\right) \left(\frac{\sqrt{x^2 + y^2}}{\sqrt{x^2 + y^2 + z^2}}\right) Simplifying this expression, we have: sin(2ϕ)=2zx2+y2x2+y2+z2\sin(2\phi) = \frac{2z\sqrt{x^2 + y^2}}{x^2 + y^2 + z^2}

Step 4: Convert the limits of integration.

The limits for ρ\rho, ϕ\phi, and θ\theta must also be converted:

  • ρ\rho ranges from 0 to 1, which implies x2+y2+z21x^2 + y^2 + z^2 \leq 1 in Cartesian coordinates. This describes the interior of a unit sphere.
  • ϕ\phi ranges from 0 to π/4\pi/4, meaning the angle between the z-axis and the radial direction ranges from 0 to π/4\pi/4. This restricts the z-coordinate such that z12z \geq \frac{1}{\sqrt{2}} when ρ=1\rho = 1.
  • θ\theta ranges from 0 to 2π2\pi, representing a full rotation in the xy-plane.

Step 5: Combine everything.

Thus, the integral in Cartesian coordinates becomes: V2z(x2+y2)(x2+y2+z2)5/2dxdydz\int_{V} \frac{2z(x^2 + y^2)}{(x^2 + y^2 + z^2)^{5/2}} \, dx\, dy\, dz where VV is the region of integration corresponding to x2+y2+z21x^2 + y^2 + z^2 \leq 1 and z12z \geq \frac{1}{\sqrt{2}}.

Would you like a detailed explanation of how to solve this integral or have any questions?

Here are 5 related questions for further understanding:

  1. How do cylindrical coordinates differ from spherical coordinates?
  2. What are the steps to converting triple integrals between different coordinate systems?
  3. How would you evaluate the integral in spherical coordinates instead?
  4. What is the physical interpretation of the coordinate transformation?
  5. How do you apply Jacobian determinants when converting between coordinate systems?

Tip: Always consider the geometric interpretation of limits in integrals when converting between coordinate systems, as it simplifies understanding the region of integration.

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Math Problem Analysis

Mathematical Concepts

Triple Integral
Cylindrical Coordinates
Cartesian Coordinates
Coordinate Transformation

Formulas

x = \rho \cos(\theta)
y = \rho \sin(\theta)
z = \rho \sin(\phi)
\rho^3 = (x^2 + y^2)^{3/2}
\sin(2\phi) = \frac{2z\sqrt{x^2 + y^2}}{x^2 + y^2 + z^2}

Theorems

Cylindrical to Cartesian Coordinate Conversion
Double Angle Identity

Suitable Grade Level

Undergraduate Mathematics