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I know how to change the intervals of an integral, for example the integral of $(\sin x)^2$ from $-\pi$ to $\pi$ is equal to $\pi\int_{-1}^1 (\sin πx)^2 \,dx$.

I find it difficult to do that in 2D. In particular i want the

$$ \int_1^2 \int_3^4 \sin(\pi x)\sin(\pi y)\, dx\, dy $$

to change intervals and become $(-1,1)$ $(-1,1)$

**Thanks for the information but i would like specifically to do a double integration on the square [-1,1]^2 . I mean that $$ \int_1^2 \int_3^4 \sin(\pi x)\sin(\pi y)\, dx\, dy $$

i want it to change to $$ \int_{-1}^1 \int_{-1}^1 $$

Marios Theodoulou
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2 Answers2

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In this case, since the integrand is just a product of functions of $x$ and $y$, you can write it as $$\int_3^4 \sin(\pi x) dx \int_1^2 \sin(\pi y) dy .$$ The linear transformations should then be straightforward.

This assumes that $dx$ being inside means that the inner integral is on $x$.

marty cohen
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  • This is correct,but I would hesitate to give this advice without explaination since this is only true by the Cauchy-Fubini multiple integral theorem because the functions in the integrands are very smooth on thier domains of definition.My initial thought would be to transform the region and integrands in terms of polar coordinates,but I think the result may be more complicated in this case. – Mathemagician1234 Oct 23 '14 at 03:13
  • Also,if these conditions are met,the order of partial integrations on the region shouldn't matter. – Mathemagician1234 Oct 23 '14 at 03:36
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In higher dimensions,we usually don't simply change intervals by multiplying by a real number,it's usually more complex then that. What we cando,which is more powerful and versatile,is try and transform the region from Cartesian to polar coordinates. In this case,though-not only would the resulting computation, using the addition and product angle formulas,be vastly more complicated,it's not necessary. Because the integrand is a sine function ,which is infinitely differentiable on the region. by the Cauchy-Fubini Theorum, we can rewrite the double integral as a product of integrals:

$$\int_3^4 \sin(\pi x) dx \int_1^2 \sin(\pi y) dy.$$

Also,by the Cauchy-Fubini Theorem,the order of integration with respect to x or y shouldn't matter.

Mathemagician1234
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