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MA 242.003
• Day 46 – March 19, 2013• Section 9.7: Spherical Coordinates• Section 12.8: Triple Integrals in Spherical Coordinates
Section 12.8Triple Integrals in Spherical Coordinates
Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.
Section 12.8Triple Integrals in Cylindrical Coordinates
Spheres
Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.
Section 12.8Triple Integrals in Cylindrical Coordinates
Spheres
Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.
Cones
To study spherical coordinates to use with triple integration we must:
1. Define spherical Coordinates (section 9.7)
1. Define spherical Coordinates (section 9.7)
2. Set up the transformation equations
To study spherical coordinates to use with triple integration we must:
1. Define spherical Coordinates (section 9.7)
2. Set up the transformation equations
To study spherical coordinates to use with triple integration we must:
3. Study the spherical coordinate Coordinate Surfaces
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
3. Study the cylindrical coordinate Coordinate Surfaces
4. Define the volume element in spherical coordinates:
To study cylindrical coordinates to use with double integration we must:
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
3. Study the cylindrical coordinate Coordinate Surfaces
4. Define the volume element in spherical coordinates:
To study cylindrical coordinates to use with double integration we must:
in cylindrical coordinates
1. Define Cylindrical Coordinates (section 9.7)
2. Set up the transformation equations
3. Study the cylindrical coordinate Coordinate Surfaces
4. Define the volume element in spherical coordinates:
To study cylindrical coordinates to use with double integration we must:
in cylindrical coordinates
in Cartesian coordinates
1. Define Spherical Coordinates
2. Set up the Transformation Equationsa. To transform integrands to spherical coordinatesb. To transform equations of boundary surfaces
2. Set up the Transformation Equationsa. To transform integrands to spherical coordinatesb. To transform equations of boundary surfaces
2. Set up the Transformation Equationsa. To transform integrands to spherical coordinatesb. To transform equations of boundary surfaces
2. Set up the Transformation Equationsa. To transform integrands to spherical coordinatesb. To transform equations of boundary surfaces
3. Study the Spherical coordinate Coordinate Surfaces
Definition: A coordinate surface (in any coordinate system) is a surface traced out by setting one coordinate constant, and then letting the other coordinates range over there possible values.
3. Spherical coordinate Coordinate Surfaces
The = constant coordinate surfaces
The = constant coordinate surfaces
3. Spherical coordinate Coordinate Surfaces
The = constant coordinate surfaces
3. Spherical coordinate Coordinate Surfaces
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
3. Spherical coordinate Coordinate Surfaces
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
A rectangular box in Cartesian coordinates
3. Spherical coordinate Coordinate Surfaces
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
A cylindrical box in cylindrical coordinates
3. Spherical coordinate Coordinate Surfaces
Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.
A spherical box in spherical coordinates
4. Define the volume element in spherical coordinates:
Section 12.8Triple Integrals in Cylindrical Coordinates
Spheres
Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.
Cones
Fubini’s Theorem in spherical coordinates
Plausibility argument: Let f(x,y,z) be continuous on the spherical box (spherical wedge) described by
Fubini’s Theorem in spherical coordinates
Plausibility argument: Let f(x,y,z) be continuous on the spherical box (spherical wedge) described by
Partitioning using spherical boxes and using the spherical volume element for each sub box we find
The following approximation of a triple Riemann sum
The following approximation of a triple Riemann sum
But this is an actual triple Riemann sum for the function
The following approximation of a triple Riemann sum
But this is an actual triple Riemann sum for the function
(Continuation of example)
(Continuation of example)
(Continuation of example)
(Continuation of example)
(Continuation of example)
