Math Problem Statement
Question content area top Part 1 Evaluate the integral Integral from 0 to 6 Integral from negative 2 pi to 0 Integral from pi divided by 4 to pi divided by 2 rho cubed sine 2 variant phi font size decreased by 3 d variant phi font size decreased by 3 d theta font size decreased by 3 d rho.
Solution
This integral involves triple integration in spherical coordinates. Let’s carefully evaluate it.
Given integral:
Step 1: Analyze the integrand and order of integration
- : This represents the radial coordinate, and it ranges from to .
- : This is the azimuthal angle, and it ranges from to .
- : This is the polar angle (or colatitude), and it ranges from to .
- The integrand is .
We’ll integrate in the following order: first , then , and finally .
Step 2: Integrate with respect to
We need to evaluate:
The antiderivative of is:
Evaluating this at the limits:
[ -\frac{1}{2} \left[ \cos\left(2 \cdot \frac{\pi}{2}\right) - \cos\left(2 \cdot \frac{\pi}{4}\right) \right] = -\frac{1}{2} \left[ \cos(\pi) - \cos\left(\frac{\pi}{2}\right) \right]. ]
Using and , this becomes:
Step 3: Integrate with respect to
Now we need to evaluate:
This simplifies to:
Step 4: Integrate with respect to
Now we integrate:
Factor out the :
The antiderivative of is:
Evaluating at the limits:
Thus:
Step 5: Final Answer
The value of the integral is:
Would you like more details on any of these steps?
Here are 5 related questions you might find interesting:
- How are spherical coordinates defined and converted to Cartesian coordinates?
- How do you determine the limits of integration in triple integrals?
- What are some common uses of spherical coordinates in physics?
- How does the integration of trigonometric functions differ from polynomial functions?
- What are alternative methods for evaluating triple integrals?
Tip: When working with integrals involving trigonometric functions, keep symmetry in mind—it can often simplify calculations!
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Math Problem Analysis
Mathematical Concepts
Calculus
Triple Integration
Spherical Coordinates
Formulas
Triple integral formula: ∭ f(ρ, θ, φ) dV in spherical coordinates
Spherical coordinates: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ)
Integration of trigonometric functions: ∫ sin(2φ) dφ
Theorems
Integration by Parts
Fundamental Theorem of Calculus
Suitable Grade Level
College Level
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