Change the order of triple integration

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Homework Help Overview

The problem involves changing the order of integration for a triple integral originally expressed as \(\int_{0}^{2}\int_{0}^{y^3}\int_{0}^{y^2}dzdxdy\). The task is to rewrite this integral with the order dydzdx.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the process of sketching the region of integration and express uncertainty about setting the limits for the variables z and y. There is a suggestion to include variables in the limits of integration for clarity.

Discussion Status

Some participants have proposed potential new limits for the integral, while others express concerns about the completeness of the problem statement. There is an acknowledgment that both the original and the rewritten integrals should represent the same volume, but uncertainty remains about how to verify this without a specific function to integrate.

Contextual Notes

Participants note that the problem may lack clarity and completeness, leading to confusion about the limits of integration and the overall approach to changing the order.

coco87
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Homework Statement


Rewrite \int_{0}^{2}\int_{0}^{y^3}\int_{0}^{y^2}dzdxdy as an integral with order dydzdx.

Homework Equations


N/A


The Attempt at a Solution


Honestly, I got as far as sketching it:
dzdxdy-graph.jpg


and after sketching it, I'm lost...

I can't figure out how to set z or y, but I'm fairly sure that x is [0,8].

could anyone possibly offer some help?

Thanks
 
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Is this all you're given in the problem? Just change the order of integration? I'm concerned that you might be omitting some information in the problem.

BTW, I've always found it to be easier to include the variable in one of the limits of integration, when I'm dealing with iterated integrals.
\int_{y = 0}^{2}\int_{x = 0}^{y^3}\int_{z = 0}^{y^2}dz~dx~dy
 
Mark44:

Thank you for your tip :)

I'm actually not leaving anything out at all. Many of these questions are vague (and rather annoying). I think I might have it, but am not sure:

\int_{0}^{8}\int_{0}^{\sqrt[3]{x}}\int_{0}^{\sqrt{z}}dydzdx. However, this seems way to easy to be true...
 
Well, you can check. both integrals represent the volume of the region, so both integrals should produce the same value.
 
Mark44:

Hmm, how would I check without a function to integrate?
 

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