Struggling with the integration part

  • Thread starter rolylane
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In summary, the given integral can be solved by using the substitution u = r^2 and evaluating the double integral with respect to u and then θ. The final answer is \sqrt{2} \left(\frac{64\pi}{3} + 16\pi\right).
  • #1
rolylane
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Hi
I've been working on a problem and I'm nearly there but I'm struggling with the integration part at the end and was hoping you might be able to help if you have the time. The original question was

[tex]\int \int (y^2 z^2 + z^2 x^2 + x^2 y^2) \: dS[/tex]
Evaluated on the region of [tex]z^2 = x^2 + y^2[/tex] between z=1 and z=2.
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Then by substituting in x=r*cos(θ) and y=r*sin(θ), and multiplying by 'r' (the Jacobian determinant), I got [tex]\sqrt{2} \int_{0}^{2\pi} \int_{1}^{2}(r^5 + r^3 \cos(\theta)^2 \sin(\theta)^2) \: dr \: d\theta[/tex]

and then I'm stuck. Any help or advice would really be appreciated
Thanks
 
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  • #2
!For this integral, you can use the substitution u = r^2. This will give us: \sqrt{2} \int_{0}^{2\pi} \int_{1}^{4}(u^2 + u \cos(\theta)^2 \sin(\theta)^2) \: du \: d\thetaIntegrating with respect to u first will give us: \sqrt{2} \int_{0}^{2\pi} \left[\frac{u^3}{3} + \frac{u^2 \cos(\theta)^2 \sin(\theta)^2}{2}\right]_1^4 \: d\thetaEvaluating this integral and multiplying by \sqrt{2} will give us the final answer: \sqrt{2} \left[\frac{64}{3} + 8\sin(\theta)^2 \cos(\theta)^2\right]_0^{2\pi}The final answer is therefore: \sqrt{2} \left(\frac{64\pi}{3} + 16\pi\right)
 
  • #3
for reaching out for help with the integration part of your problem. It looks like you're on the right track with your substitution and using the Jacobian determinant. To continue, you can use the trigonometric identity cos^2θsin^2θ = (1/4)sin^2(2θ) to simplify the integrand. Then, you can use the power rule and the fundamental theorem of calculus to evaluate the integral. Remember to also include the limits of integration based on the given region. I hope this helps, and don't hesitate to reach out if you need further assistance. Good luck!
 

1. What does "struggling with the integration part" mean?

Struggling with the integration part refers to having difficulty incorporating new or different ideas, concepts, or components into an existing system or process.

2. Why is integration important in science?

Integration is important in science because it allows for the combination of different perspectives, data, and methods to create a more comprehensive and accurate understanding of a phenomenon or problem. It also allows for the creation of more efficient and effective solutions.

3. What are some common challenges in integrating new ideas or components?

Some common challenges in integrating new ideas or components include compatibility issues, conflicting theories or data, and difficulty in finding a balance between the old and new components.

4. How can one overcome struggles with integration in science?

To overcome struggles with integration in science, it is important to have a clear understanding of the goal or problem at hand and to communicate and collaborate effectively with others. It may also be helpful to break down the integration process into smaller, manageable steps and to be open to feedback and adjustments.

5. What are some benefits of successfully integrating new ideas or components?

Successfully integrating new ideas or components can lead to a more holistic and accurate understanding of a problem, increased efficiency and effectiveness in solutions, and the potential for new discoveries and innovations. It also fosters collaboration and promotes progress in the scientific community.

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