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David E Speyer
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Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG

$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} dx dy = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$

Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG

$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$

Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG

$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} dx dy = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$

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David E Speyer
  • 156.2k
  • 14
  • 419
  • 763

Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG

$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$