Calculating Volume in the First Quadrant: y = x^4, y = \sqrt[4]{x}, and z = xy^3

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In summary, the conversation is about finding the volume of a region in the first quadrant of the x-y plane bounded by y = x^4, y = \sqrt[4]{x}, and z = xy^3. The discussion includes different approaches to solving the problem, with the conclusion that the integral cannot be done with the given limits of integration.
  • #1
benabean
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Can you verify this please?

Find the volume of the region whose base in the first quadrant of the x-y plane is bounded by [itex]y = x^4[/itex] and [itex]y = \sqrt[4]{x}[/itex], and which is bounded from above by [itex]z = xy^3[/itex]

I know it is possible to do it like so:

[itex]\left[\int_{x=0}^{1}\int_{y=0}^{\sqrt[4]{x}}xy^3 dxdy\right] - \left[\int_{x=0}^{1}\int_{y=0}^{x^4}xy^3 dxdy\right][/itex]


but can I do it such: [itex]\int_{x=y^4}^{\sqrt[4]{y}}\int_{y=x^4}^{\sqrt[4]{x}}xy^3 dxdy[/itex]

I arrive at the problem of subbing in the limits. I'm not sure if they're correct but by the looks of it to me, the limits of both integrals are dependent on the other variable so I don't know which one to do first?

thanks for your help, b.
 
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  • #2
benabean said:
but can I do it such: [itex]\int_{x=y^4}^{\sqrt[4]{y}}\int_{y=x^4}^{\sqrt[4]{x}}xy^3 dxdy[/itex]

I arrive at the problem of subbing in the limits. I'm not sure if they're correct but by the looks of it to me, the limits of both integrals are dependent on the other variable so I don't know which one to do first?

That's your clue that you can't do the integral that way. You can't integrate over y and still have a y in the limits of the x integral, since y should have been integrated out.
 
  • #3
Since the result must be a number the limits of integration of the "outer integral" must be numbers, not functions of some other variable. Once you have done the "dy" integral, there is no longer any "y" in the problem.

You could, of course, do it as
[tex]\int_{x=0}^1\int_{y= x^4}^{^4\sqrt{x}} xy^2 dydx[/tex]
 
  • #4
Thanks guys, your help is much appreciated.
 

1. Can you verify this please?

As a scientist, one of my main responsibilities is to verify scientific information and experiments. Please provide me with the specific details or claims that you would like me to verify.

2. How do you verify scientific information?

As a scientist, I use the scientific method to verify information. This involves developing a hypothesis, designing and conducting experiments, analyzing data, and drawing conclusions based on evidence.

3. Can you explain the process of verification in science?

The process of verification in science involves conducting experiments and collecting data to test a hypothesis. This data is then analyzed and compared to existing theories and research to determine its validity.

4. What is the importance of verification in science?

Verification is crucial in science as it ensures the accuracy and reliability of scientific information. It also allows for the replication of experiments and the building of new knowledge on existing theories.

5. How do you ensure the accuracy of your verification process?

To ensure accuracy, scientists use rigorous experimental designs, control groups, and statistical analysis. Experiments are also peer-reviewed and replicated by other scientists to validate the results.

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