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In the book of Giasser Cosmic Rays and particles. It says that the geometrical acceptance of two parallel plates of area $A1$ and $A2$, separated by a distance $d$ is:

$$A = \Delta \Omega \int_{0}^{2\pi} d\varphi_{1} \int_{0}^{r_{max}}r_{1}dr_{1}\int_{0}^{2\pi}d\varphi_{2}\int_{\xi }^{1}cos\left ( \theta \right )d \ cos\ \theta$$

But how is the last integral evaluated, I have never seen that form. Is it considering that the $cos \ \theta$ is the variable of integration?

Alessio K
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Las Des
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  • Why no distance $d$ appears anywhere in the expression? – Narasimham Sep 26 '21 at 17:58
  • Apparently the term $\xi$ is $\xi \left ( r_{1},\varphi_{1},\varphi_{2} \right )$ is a term of the angular integral, that is determined by the shape of the upper plane, but there is no explicit form on how this function is obtained; could by geometric construction. – Las Des Sep 26 '21 at 18:16

1 Answers1

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Yes, it is a change of variable. So we have

$$\int\cos(\theta)d\cos(\theta)=\frac{\cos^2(\theta)}{2}+C$$

since $\int\cos(\theta)d\cos(\theta)$ is a function such that its derivative wrt $\cos(\theta)$ is $\cos(\theta)$. See Integration of a function with respect to another function.

Alessio K
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