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:

062π0π4π2ρ3sin(2φ)dφdθdρ.\int_0^6 \int_{-2\pi}^0 \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \rho^3 \sin(2\varphi) \, d\varphi \, d\theta \, d\rho.

Step 1: Analyze the integrand and order of integration

  • ρ\rho: This represents the radial coordinate, and it ranges from 00 to 66.
  • θ\theta: This is the azimuthal angle, and it ranges from 2π-2\pi to 00.
  • φ\varphi: This is the polar angle (or colatitude), and it ranges from π4\frac{\pi}{4} to π2\frac{\pi}{2}.
  • The integrand is ρ3sin(2φ)\rho^3 \sin(2\varphi).

We’ll integrate in the following order: first dφd\varphi, then dθd\theta, and finally dρd\rho.


Step 2: Integrate with respect to φ\varphi

We need to evaluate:

π4π2sin(2φ)dφ.\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sin(2\varphi) \, d\varphi.

The antiderivative of sin(2φ)\sin(2\varphi) is:

12cos(2φ).-\frac{1}{2} \cos(2\varphi).

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 cos(π)=1\cos(\pi) = -1 and cos(π/2)=0\cos(\pi/2) = 0, this becomes:

12[10]=12.-\frac{1}{2} \left[ -1 - 0 \right] = \frac{1}{2}.


Step 3: Integrate with respect to θ\theta

Now we need to evaluate:

2π012dθ.\int_{-2\pi}^0 \frac{1}{2} \, d\theta.

This simplifies to:

12[θ]2π0=12(0(2π))=12(2π)=π.\frac{1}{2} \left[ \theta \right]_{-2\pi}^0 = \frac{1}{2} \left( 0 - (-2\pi) \right) = \frac{1}{2} (2\pi) = \pi.


Step 4: Integrate with respect to ρ\rho

Now we integrate:

06ρ3πdρ.\int_0^6 \rho^3 \pi \, d\rho.

Factor out the π\pi:

π06ρ3dρ.\pi \int_0^6 \rho^3 \, d\rho.

The antiderivative of ρ3\rho^3 is:

ρ44.\frac{\rho^4}{4}.

Evaluating at the limits:

14[6404]=14(1296)=324.\frac{1}{4} \left[ 6^4 - 0^4 \right] = \frac{1}{4} (1296) = 324.

Thus:

π324=324π.\pi \cdot 324 = 324\pi.


Step 5: Final Answer

The value of the integral is:

324π.324\pi.


Would you like more details on any of these steps?


Here are 5 related questions you might find interesting:

  1. How are spherical coordinates defined and converted to Cartesian coordinates?
  2. How do you determine the limits of integration in triple integrals?
  3. What are some common uses of spherical coordinates in physics?
  4. How does the integration of trigonometric functions differ from polynomial functions?
  5. 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