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xartaan
 2 years ago
Check my work on changing the order of integration?
\[\int\limits_{0}^{1} \int\limits_{0}^{x^3} e^{x}\sin(y)dydx\]
xartaan
 2 years ago
Check my work on changing the order of integration? \[\int\limits_{0}^{1} \int\limits_{0}^{x^3} e^{x}\sin(y)dydx\]

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xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{0}^{1} \int\limits_{0}^{y^{1/3}} e^{x}\sin(y)dxdy\]

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0I just drew a picture, and since I had y=x^3 I figured solving for x gives y^(1/3) .. Am I close, or way off?

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0I guess it didn't copy well in the initial post, here is the original ntegral: \[\int\limits_{0}^{1} \int\limits_{0}^{x^3} e^{x}\sin(y)dydx\]

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0Ok cool, that is what my sketch looked like as well. So I did it right?

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1\[\huge \large \int\limits_{y=0}^1\quad\int\limits_{x=y^{1/3}}^1\quad e^x \;\sin y \;dx\;dy\] Hmm like this I think?

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0Isn't x going from 0 to y^1/3 not from 1? I guess I am missing something

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1362707659927:dwSo this is what our original setup looks like. yes? Y is going from y=0 up to the function.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1We're switching the limits, so now our X will relate to the function.dw:1362707796144:dw

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0oooooooh, I think I see! so x goes from the curve, to the line x=1

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0Ok, last question then. Looking at this, how do I decide wich is at the top of the integral and which is on bottom? My guess is y^1/3 is on top because it is father left than the line x=1, so its smaller values?

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0Wait, smaller value would put it on the bottom.

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1We always integrate \(\large \text{left to right}\) and \(\large \text{bottom to top}\). Which in turn will always be smallest to largest, like you said. :)

xartaan
 2 years ago
Best ResponseYou've already chosen the best response.0<3 Thank you so much for clearing this up for me!

zepdrix
 2 years ago
Best ResponseYou've already chosen the best response.1We can see from the picture, that x=y^(1/3) is SMALLER that x=1 in the given interval right? :) Yay glad I could help.
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