Math Problem Statement
Solution
The problem asks us to find the volume of a sphere with radius using a triple integral. Let’s solve it.
Step-by-Step Solution:
We use spherical coordinates for simplicity in evaluating the triple integral. In spherical coordinates:
- The Jacobian determinant (volume element) is .
The sphere's radius is , so the region is bounded by:
The volume of the sphere is given by:
Step 1: Evaluate the innermost integral ():
Step 2: Evaluate the second integral ():
Step 3: Evaluate the outermost integral ():
Final Step: Multiply the results:
Final Answer:
The volume of the sphere is:
Would you like me to break down any of the steps further or provide additional explanations?
Related Questions:
- How is the Jacobian determinant derived in spherical coordinates?
- What are the limits for , , and in spherical coordinates for a sphere?
- Can you explain the difference between Cartesian and spherical coordinate systems?
- How would the solution change for a hemisphere instead of a sphere?
- What is the physical significance of the volume element ?
Tip:
When solving triple integrals in spherical coordinates, always carefully define the limits of integration and include the Jacobian determinant.
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Math Problem Analysis
Mathematical Concepts
Triple Integrals
Spherical Coordinates
Volume of a Sphere
Formulas
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
Jacobian determinant = ρ² sin(φ) dρ dφ dθ
Volume integral: V = ∫∫∫ ρ² sin(φ) dρ dφ dθ
Theorems
Triple Integral Theorem
Conversion to Spherical Coordinates
Suitable Grade Level
Undergraduate (Calculus III or Advanced Mathematics)
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