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Triple Integral in Spherical Coordinates

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Spherical Coordinates

𝜌 = distance |OP| (𝜌≥0) 𝜃 = angle of the projection of the point into the 𝑥𝑦-plane measured from the positive x-axis (0≤𝜃≤2𝜋 ) ϕ = angle |OP| makes with the positive z- axis (0 ≤ ϕ ≤ π; if ϕ > π/2 the point has a negative z-coordinate) Relationship with cartesian: with thus Also:

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Spherical Coordinates

Basic graphs in spherical coordinates: 𝜌=𝑐 represents a sphere of radius 𝑐 ( 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑐 2 in cartesian) ϕ = c represents a cone 𝜃=𝑐 represents a vertical plane

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Spherical Coordinates

Plot the point and convert to cartesian: Change from rectangular to spherical: The projection of the point in the 𝑥𝑦-plane is in the first quadrant

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Spherical Coordinates Example 1

Write the equation of the cone 𝑧 2 = 𝑥 2 + 𝑦 2 in spherical coordinates. Recall that Also, 𝑥 2 + 𝑦 2 = 𝑟 2 and Substituting into the equation of the cone, yields: Simplifying: Thus is the equation of the top half of the cone. is the equation of the bottom half of the cone.

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Triple Integrals in spherical coordinates

The volume element is Theorem: Change of coordinates where E is the spherical wedge given by

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Triple Integrals in Spherical Coordinates – Example 4

Evaluate where E is the hemispherical region that lies above the 𝑥𝑦-plane and below the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 =9 Spherical coordinates: hemisphere

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Triple Integrals in Spherical Coordinates – Example 5

Find the volume of the solid that lies within the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 =49, above the 𝑥𝑦-plane and outside the cone 𝑧=4 𝑥 2 + 𝑦 2 We need to determine the angle that describes the cone in spherical coordinates.

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Triple Integrals in Spherical Coordinates – Example 6

Evaluate the integral by changing to spherical coordinates The solid is bounded below by the cone 𝑧= 𝑥 2 + 𝑦 2 and above by the hemisphere 𝑧= 2− 𝑥 2 − 𝑦 2 . The radius of the hemisphere is 2 The projection of the solid in the 𝑥𝑦-plane is the quarter of the disk 𝑥 2 + 𝑦 2 ≤1 in the first quadrant.

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Triple Integrals in Spherical Coordinates Example 6 continued

the cone has equation the sphere has equation 𝜌= 2 The limits of integration are

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