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$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like?

You mean something like this? ( I made mess)

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like?

So $$ \int_{y=x}^{2x}\ (x+2y)\ dx = \int_{y=x}^{2x} x dx + \int_{y=x}^{2x}\ 2y dx = \frac{1}2 [x^{2}] + 2y \int_{y=x}^{2x}\ 1 dx = \frac{1}{2}(2x)^{2}-y^{2} + 2y(2x-y) $$

and $$ \int_{x=2}^3 [ \frac{1}{2}(2x)^{2}-y^{2} + 2y(2x-y) ]dy = \frac{1}{2}\int_{x=2}^3 (2x)^{2} dy - \int_{x=2}^3 y^{2} dy + \int_{x=2}^3 4xy dy - \int_{x=2}^3 2y^{2} dy = \frac{1}{2} (2x)^{2} \int_{x=2}^3 1 dy - [\ \frac{1}{3}y^{3}] + 4x[\frac{1}{2}y^{2}] - 2[\frac{1}{3}y^{3}]= \frac{1}{2} (2x)^{2}... $$ and I do not know.

Tunk-Fey
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Mango
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  • You need to write $D={(x,y)\in \mathbb R^2\colon a\leq x\leq b\land f_1(x)\leq y\leq f_2(x)}$ or $D={(x,y)\in \mathbb R^2\colon c\leq y\leq d\land g_1(y)\leq x\leq g_2(y)}$, with $ab,c,d\in \mathbb R$ and for some well-behaved functions $f_1,f_2,g_1,g_2$. Then the integral becomes, for instance in the first case, $\displaystyle \int \limits_a^b\int \limits_{f_1(x)}^{f_2(x)}(x+2y),\mathrm dy,\mathrm dx$. – Git Gud May 26 '14 at 20:34
  • No. You misinterpret my answer. You should evaluate $\displaystyle\int_{y=x}^{2x}(x+2y)\ dy$ first. See the comments below my answer. – Tunk-Fey May 26 '14 at 20:54

1 Answers1

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Hint : $$ \iint_D\ (x+2y)\ dx\,dy=\int_{x=2}^3\int_{y=x}^{2x}\ (x+2y)\ dx\,dy. $$

Tunk-Fey
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  • See this plot to visualize the region. – Tunk-Fey May 26 '14 at 20:23
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    You mispelled 'answer'. – Git Gud May 26 '14 at 20:23
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    It should be $\mathrm dy,\mathrm dx$. – Git Gud May 26 '14 at 20:36
  • @GitGud My Prof says it doesn't matter as long as we include the variable in the limit of integrals so we know what we should evaluate first. For example $$ \int_{x=2}^3\int_{y=x}^{2x}\ (x+2y)\ dx,dy=\int_{2}^3\int_{x}^{2x}\ (x+2y)\ dy,dx $$ – Tunk-Fey May 26 '14 at 20:41
  • I see. I didn't know about that notation. – Git Gud May 26 '14 at 20:43
  • @GitGud Similar notation is also used in textbook Schaum’s Outline Series: Probability and Statistics Third Edition by Murray R. Spiegel and John J. Schiller. – Tunk-Fey May 26 '14 at 20:48
  • Thank you :) I didn't see that, i calculate it and now is good. answer is $25 \frac{1}{3}$ – Mango May 26 '14 at 21:14
  • @GitGud, if the limits of one axis depend on the other, the itereated integrals should of that order. I agree with you, it could be made wrong if he evaluates it in the wrong order. I see this confusion in many of the answers from other part of the world. – Satish Ramanathan Jun 17 '14 at 12:48