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Reversing Order of Integration: Double Integral Evaluation
- Thread starter BrownianMan
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The summary of the conversation is: "In summary, to evaluate the integral by reversing the order of integration, the first step is to sketch a graph of the region over which integration is taking place. Then, for the inner integral, x ranges from cuberoot(y) to 2, and for the outer integral, y ranges from 0 to 8. The iterated integral with the opposite order will have inner limits involving two functions of y and outer limits involving two x values."
![gif.latex?\int_{0}^{8}\int_{\sqrt[3]{y}}^{{2}}7e^{x^4}dxdy.gif](/data/attachments/121/121114-9c5994283ab8624f2d7259db665996dc.jpg){kind=small}
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Mark44
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The first step in this type of problem is to sketch a graph of the region over which integration is taking place. For the inner integral, x ranges from cuberoot(y) to 2. For the outer integral, y ranges from 0 to 8. Sketch the graphs of x = y1/3 and x = 2, and then sketch the graphs of y = 0 and y = 8. For the iterated integral with the opposite order, the inner integration limits will involve two functions of y, and the outer integration limits will involve two x values.BrownianMan said:Evaluate the integral by reversing the order of integration.
![gif.latex?\int_{0}^{8}\int_{\sqrt[3]{y}}^{{2}}7e^{x^4}dxdy.gif](https://www.physicsforums.com/attachments/gif-latex-int_-0-8-int_-sqrt-3-y-2-7e-x-4-dxdy-gif.138033/ "gif.latex?\int_{0}^{8}\int_{\sqrt[3]{y}}^{{2}}7e^{x^4}dxdy.gif")
I'm not exactly sure how to approach this problem. Any help would be appreciated!
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BrownianMan
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Ok, so would the answer be
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Mark44
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Yep.
Related to Reversing Order of Integration: Double Integral Evaluation
1. What is the purpose of reversing the order of integration in a double integral?
Reversing the order of integration allows us to more easily evaluate a double integral by changing the order in which we integrate with respect to the two variables. This can make the integral simpler and sometimes even possible to solve analytically.
2. How do I know when to reverse the order of integration in a double integral?
The order of integration should be reversed when the original integral is difficult to evaluate or when the limits of integration are easier to express in the opposite order. This may also be necessary when using certain integration techniques, such as substitution or integration by parts.
3. What are the steps for reversing the order of integration in a double integral?
The first step is to draw a graph of the region of integration and identify the limits of integration for each variable. Then, write the integral in the opposite order, using the new limits of integration. Finally, evaluate the integral using the new order of integration.
4. Can reversing the order of integration change the value of a double integral?
No, reversing the order of integration does not change the value of a double integral. It simply changes the way in which the integral is evaluated. However, it can make a previously unsolvable integral possible to solve or make the evaluation process more efficient.
5. Are there any limitations to reversing the order of integration in a double integral?
Yes, there are certain scenarios where reversing the order of integration may not be possible or may not lead to a simpler integral. This can occur when the region of integration is not well-defined or when the limits of integration are too complex to be easily expressed in the opposite order.