Computing $\int_{0}^{1}\int_{0}^{1}\int_{\max\{x,y\}}^{1} e^{z^3} dz dx dy$

Asked Jun 25 '15 at 07:33

Active Jun 25 '15 at 09:43

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I am trying to compute

$$\int_{0}^{1}\int_{0}^{1}\int_{\max\{x,y\}}^{1} e^{z^3} dz dx dy$$

What I have done is to reverse the order of integration; so I did the integration with respect to $x$ and $y$ first.

I get this in the end:

$$\int_{\max\{x,y\}}^{1} e^{z^3} dz$$ which doesn't solve my problem.

Perhaps I am missing something. I'm not sure which values of $x$ and $y$ I have to look at (to take the maximum).

Note: This is not a homework problem - it's a practice question from my lectures which is not graded.

edited Jun 25 '15 at 09:43

Did

asked Jun 25 '15 at 07:33

AstroInt

Maybe writing $$\int_0^1\int_0^1\int_{\max(x,y)}^1(\bullet),dzdydx=\int_0^1\int_x^1\int_y^1( \bullet ),dzdydx+\int_0^1\int_0^x\int_x^1(\bullet),dzdydx$$ will help you understand what's wrong with what you've done. – Jun 25 '15 at 07:48
3

Note that $z>\max{x,y}$ if and only if $x<z$ and $y<z$, hence $$\int_{0}^{1}\int_{0}^{1}\int_{\max{x,y}}^{1} e^{z^3} dz dx dy= \int_{0}^{1}e^{z^3}\left(\int_{0}^{z}\int_{0}^{z} dx dy\right) dz= \int_{0}^{1}z^2e^{z^3}dz=\ldots$$ – Did Jun 25 '15 at 09:40

0 Answers0