Which of the two answers is correct?

In summary: Why do the solutions have different units ? Junk.In summary, the conversation is about two different solutions to a problem involving the calculation of a volume. Solution 1 computes the integral of (x^2+y^2+z^2) while Solution 2 computes the integral of 1. After a closer look, it is determined that Solution 1 does not actually calculate the volume, but rather a different integral. The conversation ends with a request for further confirmation from other members.
  • #1
WMDhamnekar
MHB
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TL;DR Summary
Finding the volume above cone ##z= a\sqrt{x^2 + y^2}## and inside the sphere ## x^2 + y^2 + z^2 = b^2##
1654357120654.png


Solution 1:
1654357151592.png

The answer is ## \frac{2b^5\pi}{5} \times \left(1 -\frac{a}{\sqrt{1+a^2}}\right)##

Solution 2:
1654357300119.png


I want to decide which answer is correct? Would you help me in this task?
 
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  • #2
I am with solution 2, easier to understand for me and I find it correct
 
  • #3
After a closer look at Sol1. it computes not the volume V but the integral $$\iiint_V (x^2+y^2+z^2)dxdydz$$. The result of Sol1. would be the same as Sol2. if instead it would calculate $$\iiint_V 1 dxdydz=\int_0^{2\pi}\int_0^a\int_0^b \rho^2\sin\phi d\rho d\phi d\theta$$
 
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  • #4
Delta2 said:
After a closer look at Sol1. it computes not the volume V but the integral $$\iiint_V (x^2+y^2+z^2)dxdydz$$. The result of Sol1. would be the same as Sol2. if instead it would calculate $$\iiint_V 1 dxdydz=\int_0^{2\pi}\int_0^a\int_0^b \rho^2\sin\phi d\rho d\phi d\theta$$
Thanks for your scrutiny of both the solutions. But I want more confirmation from other elite members.
 
  • #5
WMDhamnekar said:
Thanks for your scrutiny of both the solutions. But I want more confirmation from other elite members.
Well ok, just to emphasize that the two solutions don't compute the same thing

We have a region V (as the region between the surface of a cone and a sphere).

Sol 1 computes ##\iiint_V (x^2+y^2+z^2)dxdydz##
Sol 2 computes ##\iiint_V 1 dxdydz##
 
  • #6
Yes. Why do the solutions have different units ? Junk.
 
  • #7
hutchphd said:
Yes. Why do the solutions have different units ? Junk.
Check post #5.
 
  • #8
Yes.
 
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1. What is the process for determining which answer is correct?

The process for determining which answer is correct can vary depending on the question and the type of scientific research being conducted. Generally, scientists use the scientific method which involves making observations, forming a hypothesis, conducting experiments, and analyzing data to come to a conclusion.

2. How do you know if an answer is correct in science?

In science, an answer is considered correct if it is supported by evidence and can be replicated by other scientists. This means that the results of experiments or studies have been consistently observed and can be verified by others.

3. Can there be more than one correct answer in science?

Yes, there can be more than one correct answer in science. This is because scientific research is constantly evolving and new evidence can lead to different conclusions. Additionally, some questions may have multiple valid interpretations or solutions.

4. How do you handle conflicting answers in science?

When there are conflicting answers in science, scientists will often conduct further research and experiments to gather more evidence and come to a consensus. It is important to consider the quality and reliability of the evidence supporting each answer before making a decision.

5. Is there always a right or wrong answer in science?

In science, the concept of right or wrong is not always applicable. Instead, there are answers that are supported by evidence and those that are not. It is important for scientists to constantly question and challenge their own findings and interpretations in order to continue advancing knowledge and understanding in their field.

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