# Questions tagged [change-of-variable]

This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals

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### Confusion about calculating integral with $d(-x)$

I'm wondering if $\int_{a}^{b}g(z)d(-z)=-\int_{a}^{b}g(z)dz$. On the one hand, intuitively this seems true as this looks like we are putting a negative sign on the $\Delta z$ in the Rieman sum, so it ...
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### Correct Notation for change of variable

I am trying to clearing distinguish these cases below that have different meaning in the way the variables are used, yet the notation is similar. What are the preferred approaches to not have the ...
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### Probability density function of $X = \frac{P}{\cos(\theta)} - y\tan(\theta)$.

Assume that we have two independent random variables: $P$ is distributed on $(0,1)$ with $f_{P}(p) = 2p$. $\theta$ is uniformly distributed on $(0,2\pi)$. We then define $X,Y$ such that \begin{align*...
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### Change of coordinates problem

Consider the region D defined by $1 \leq x^2-y^2 \leq 4$ and $0 \leq y \leq \frac{3x}{5}$. In the problem, set up an integral to compute $\int\int_{D} e^{x^2-y^2} dA$. Consider the change of ...
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### A Double Integral Substitution

I am asked to compute the integral $$\int_\Omega xye^{x^2-y^2}dxdy$$ over the domain $\Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\}.$ After splitting the domain and a messy ...
### Distribution of $Z = \sin(X) \sin(Y)$ where $X$ and $Y$ are independent and uniform in $[-\pi,\pi]$?
Consider two random variables $X$ and $Y$ that are independent and uniformly distributed over a period, say $[-\pi,\pi]$. Which is the PDF (or the CDF if you prefer) of $Z = \sin(X) \sin(Y)$? This ...