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2 votes

How can I calculate $\int_{R^2}\; \exp (-x^2-xy-y^2)\,dxdy$ this Riemann Integral.

If $M=\left[ \begin{array}{cc}1&1/2\\1/2&1\end{array}\right]$ we have $(x,y)M\left( \begin{array}{c}x\\y\end{array}\right)=x^2+xy+y^2.$ Writing $ M=U^{-1}\mathrm{diag}(1/2,3/2)U$ where $U$ is ...
Letac Gérard's user avatar
2 votes

How can I calculate $\int_{R^2}\; \exp (-x^2-xy-y^2)\,dxdy$ this Riemann Integral.

Using polar coordinates: \begin{align} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-x^2-xy-y^2}dxdy & =\int_{0}^{2\pi}\int_{0}^{+\infty} re^{-r^2-r^2\cos\theta\sin\theta}\ \ drd\theta \\ &...
Morten's user avatar
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1 vote
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Book recommendation for an author that would treat Riemann integrability when $f$ is not necessarily bounded

Having a primitive doesn't imply being Riemann integrable. The theorem you are perhaps referring to states if $f$ has a primitive $F$ in $[a,b]$ and also is Riemann integrable then $\int_a^b f=F(b)-F(...
Mark's user avatar
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