# Help me to remember changing the order of integration

**URL:** <https://boards.straightdope.com/t/help-me-to-remember-changing-the-order-of-integration/781049>\
**Category:** Factual Questions\
**Created:** [February 28, 2017, 6:15pm UTC](https://boards.straightdope.com/t/help-me-to-remember-changing-the-order-of-integration/781049 "2017-02-28T18:15:31Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Saffer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saffer/32/6440_2.png) [@Saffer](https://boards.straightdope.com/u/Saffer)\
**Post date:** [February 28, 2017, 6:15pm UTC](https://boards.straightdope.com/t/help-me-to-remember-changing-the-order-of-integration/781049/1 "2017-02-28T18:15:31Z")

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I’m trying to remember how to do this and I am looking at a problem. The details of the function are not germane to this question, but the limits of integration are.

To clarify notation, let int(a,b)f(x)dx represent an integral of the function f(x) evaluated from a to b.

The problem I have looks like this  
int(0,c)[int(0,y)f(x,y)dx]dy  
and I want to swap the order of integration.

If I draw the domain on a piece of paper, the inner integral goes from x = 0, to x = y and the outer integral goes from 0 to c. So it is a triangle with the left side being the y axis from x = 0 to c, the top being parallel with the x axis at y = c, and the other line being x = y

The question is: how would this be written when changing the order of integration?

It seems to me that the inner integral becomes  
int(x,c)f(x,y)dy  
but the outer integral seems a bit off to me.  
i get  
int(0,c)[int(x,c)f(x,y)dy]dx

Is it OK that the constant c appears twice now?

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**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\
**Post date:** [February 28, 2017, 10:30pm UTC](https://boards.straightdope.com/t/help-me-to-remember-changing-the-order-of-integration/781049/2 "2017-02-28T22:30:22Z")

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You have it right, and there’s no issue with _c_ appearing twice. (Think about what would happen if you were talking about the whole square with 0 ≤ _x_ ≤ _c_ and 0 ≤ _y_ ≤ _c_.) You’re probably doing this already, but the best way I know of to think about it is to think about horizontal lines in the domain for the original integral and vertical lines for the second integral.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [March 1, 2017, 2:34pm UTC](https://boards.straightdope.com/t/help-me-to-remember-changing-the-order-of-integration/781049/3 "2017-03-01T14:34:41Z")

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Agreed. You have it right.

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**Author:** ![Saffer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saffer/32/6440_2.png) [@Saffer](https://boards.straightdope.com/u/Saffer)\
**Post date:** [March 1, 2017, 4:10pm UTC](https://boards.straightdope.com/t/help-me-to-remember-changing-the-order-of-integration/781049/4 "2017-03-01T16:10:01Z")

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Ok, thanks both!  
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