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I'm a Computer Engineer Students, and Try to get familiar with Integral, but I couldn't get the point with three form of $x$ and $y$ variables. How we can easily get the point for following equation to solve it easily?

$1) E[x]=\int_{-1}^1 \int_0^1 xdxdy $

$2) E[x]=\int_{-1}^1 \int_0^1 ydxdy $

$3) E[x]=\int_{-1}^1 \int_0^1 xydxdy $

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    Have you tried? – 2017-01-17
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    This is an example in my notes, but I confused to understanding with variable changes. @Pythagoricus – 2017-01-17

1 Answers 1

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I am going to show you the second one, and you can try the others yourself.

A double integral is an integral of an integral. The inner one is in terms of the first of the $\mathrm{d}...$ variables, the outer - in terms of the second. For you this means $$ \begin{split} \int_{-1}^1 \int_0^1 y \mathrm{d}x \mathrm{d}y &= \int_{y=-1}^{y=1} \int_{x=0}^{x=1} y \mathrm{d}x \mathrm{d}y \\ &= \int_{y=-1}^{y=1} \left(\int_{x=0}^{x=1} y \mathrm{d}x\right) \mathrm{d}y \\ &= \int_{y=-1}^{y=1} \left(\int_{x=0}^{x=1} \mathrm{d}x\right) y \mathrm{d}y \\ &= \int_{y=-1}^{y=1} \left(1-0\right) y \mathrm{d}y \\ &= \int_{y=-1}^{y=1} y \mathrm{d}y \\ &= \left. \frac{y^2}{2} \right|_{y=-1}^{y=1} \\ &= 0. \end{split} $$

HINT FOR #3

$$ \begin{split} \int_{-1}^1 \int_0^1 y \mathrm{d}x \mathrm{d}y &= \int_{y=-1}^{y=1} \int_{x=0}^{x=1} xy \mathrm{d}x \mathrm{d}y \\ &= \int_{y=-1}^{y=1} \left(\int_{x=0}^{x=1} xy \mathrm{d}x\right) \mathrm{d}y \\ &= \int_{y=-1}^{y=1} \left(\int_{x=0}^{x=1} x \mathrm{d}x\right) y \mathrm{d}y \end{split} $$

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    Yeah very nice, your answer is very shorter than answer in my notes, but one problem I couldn't get the point for the third one. – 2017-01-17
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    @MaryamPanahi show me what you can do with the third one and i will post further hints here so you can finish it off yourself. – 2017-01-17
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    @MaryamPanahi please post here what you are able to do yourself. I would prefer not to do assignments for you without seeing any effort on your part – 2017-01-17
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    I think we can convert it to $\int_{-1}^1ydy(Z)$ and $Z$ is integral over x. but no idea. – 2017-01-17
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    Would you please add a hint for third one? – 2017-01-18
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    @MaryamPanahi please see the update, can you finish it now? – 2017-01-18
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    I get the point thanks. I Approve it. but I think it's better to complete it for others when read this question as an education purpose. thanks for your nice answer. – 2017-01-19