Express ∫∫∫_S xyz dV in spherical coordinates, where S is the positive octant of the sphere of radius a, (where a > 0 is constant).
Understand the Problem
The question is asking to express the multiple integral of the function xyz over a specified volume in spherical coordinates, specifically for the positive octant of a sphere of radius a. The goal is to convert the given integral into spherical coordinates and establish the limits of integration.
Answer
$$ \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_0^a \rho^4 \sin^3 \phi \cos \theta \sin \theta \cos \phi \, d\rho \, d\phi \, d\theta $$
Answer for screen readers
The integral expressed in spherical coordinates is:
$$ \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_0^a \rho^4 \sin^3 \phi \cos \theta \sin \theta \cos \phi , d\rho , d\phi , d\theta $$
Steps to Solve
- Convert Cartesian Coordinates to Spherical Coordinates
In spherical coordinates, the relationships are defined as follows:
$$ x = \rho \sin \phi \cos \theta $$ $$ y = \rho \sin \phi \sin \theta $$ $$ z = \rho \cos \phi $$
where ( \rho ) is the radius, ( \phi ) is the polar angle, and ( \theta ) is the azimuthal angle.
- Determine the Volume Element in Spherical Coordinates
The volume element ( dV ) in spherical coordinates is given by:
$$ dV = \rho^2 \sin \phi , d\rho , d\phi , d\theta $$
- Establish the Limits of Integration
Since the integral is over the positive octant of the sphere of radius ( a ):
- ( \rho ) ranges from ( 0 ) to ( a )
- ( \phi ) ranges from ( 0 ) to ( \frac{\pi}{2} )
- ( \theta ) ranges from ( 0 ) to ( \frac{\pi}{2} )
- Set Up the Integral in Spherical Coordinates
Now, substitute ( xyz ) in terms of spherical coordinates and the volume element:
$$ \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_0^a \left( \rho \sin \phi \cos \theta \right) \left( \rho \sin \phi \sin \theta \right) \left( \rho \cos \phi \right) \rho^2 \sin \phi , d\rho , d\phi , d\theta $$
- Combine the Expressions
Combine the expressions to form the complete integral:
$$ \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_0^a \rho^4 \sin^3 \phi \cos \theta \sin \theta \cos \phi , d\rho , d\phi , d\theta $$
The integral expressed in spherical coordinates is:
$$ \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_0^a \rho^4 \sin^3 \phi \cos \theta \sin \theta \cos \phi , d\rho , d\phi , d\theta $$
More Information
This integral calculates the volume integral of the function ( xyz ) over the specified volume in spherical coordinates. Converting to spherical coordinates simplifies the evaluation of integrals in three-dimensional spaces.
Tips
- Forgetting to adjust the limits of integration appropriately for each variable in spherical coordinates.
- Not including the Jacobian (( \rho^2 \sin \phi )) in the volume element correctly.
- Misidentifying the ranges for ( \phi ) and ( \theta ).
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