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 are related to Cartesian coordinates through the following relations:
Here, represents the radial distance in the xy-plane, is the angle between the radius and the z-axis (the "polar angle"), and is the azimuthal angle (rotation in the xy-plane).
Step 2: Rewriting the integrand.
The given integral is: We want to express the integrand and the limits of integration in Cartesian coordinates.
Step 3: Convert the integrand.
In Cartesian coordinates:
- , so becomes .
- The function depends on , which is the angle between and the z-axis. In Cartesian coordinates, we can express through the relation: To express , we use the double-angle identity: where: Hence, we get: Simplifying this expression, we have:
Step 4: Convert the limits of integration.
The limits for , , and must also be converted:
- ranges from 0 to 1, which implies in Cartesian coordinates. This describes the interior of a unit sphere.
- ranges from 0 to , meaning the angle between the z-axis and the radial direction ranges from 0 to . This restricts the z-coordinate such that when .
- ranges from 0 to , representing a full rotation in the xy-plane.
Step 5: Combine everything.
Thus, the integral in Cartesian coordinates becomes: where is the region of integration corresponding to and .
Would you like a detailed explanation of how to solve this integral or have any questions?
Here are 5 related questions for further understanding:
- How do cylindrical coordinates differ from spherical coordinates?
- What are the steps to converting triple integrals between different coordinate systems?
- How would you evaluate the integral in spherical coordinates instead?
- What is the physical interpretation of the coordinate transformation?
- 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
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