# Questions tagged [change-of-variable]

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

368 questions
41 views

### Laplace Transform: Indeterminate Form in Definite Integral Change of Variables Calculation

I was trying to find the Laplace transform of $e^{3t}$: $$\int^\infty_0 e^{3t}e^{-st} \ dt = \int_0^\infty e^{3t - st} \ dt = \lim_{x \to \infty}\int_0^x e^{3t - st} \ dt$$ So if we then attempt to ...
9 views

### change of variables with limits

If I have the following function: $$1 = \int_0^{R_M}g_s(R)R^{-1}dR$$ where $x_M = R_M/\sigma_0$, and $x = R/\sigma_0$ how would I perform sub. of variables on the limits? I know I would have ...
27 views

25 views

### Change Variables in a Multiple Integral

Let $D$ be the region that's bound by $y=x^2, y=2x^2, x=y^2, x=3y^2$. $D$ corresponds to the region $E$ where $u=\frac{x^2}{y}$ and $v=\frac{y^2}{x}$. Sketch $D$ and $E$ Find the solution ...
16 views

88 views

20 views

31 views

### Change of variables for an integral

I have an integral where I need to change variables. The integral has the form, $\int_0^x f(x,t) dt$ . I change variables/rescale by setting $\tilde{t}=xt$, which means $d\tilde{t}=xdt$. Would the ...
75 views

### Rudin's proof of change of variables.

When I read Rudin's proof of change of variables, I have a problem underlined in red below: I don't understand why (31) is true when $T$ is a primitive $C'$-mapping. I know that a primitive mapping ...
46 views

### Show that $(X,Y)$ has bivariate normal distribution when $X = Z_{1}$ and $Y = Z_{1} + Z_{2}$, where $Z_{i}\sim\mathcal{N}(0,1)$

Assume that $Z_{1}$ and $Z_{2}$ are independent standard normally distributed random variables. Show that $(X,Y)$ has bivariate normal distribution when $X = Z_{1}$ and $Y = Z_{1} + Z_{2}$. MY ...
28 views

### Transformation of continuous random variables: solution verification

Assume $X$ to be a random variable whose probability density function is given by \begin{align*} f_{X}(x) = \begin{cases} \displaystyle\frac{3x^{2}}{2} & \text{if}\,\,\,-1\leq x \leq 1\\\\ 0 &...
21 views

42 views

26 views

### Formula to calculate change in distance to destination or origin of a straight-line path of travel

I am writing an application that consumes GPS data - and I am trying to calculate direction traveled based on a change in distance to the destination and origin. Assume that I have a straight path of ...
64 views

### Changing order of integration:$\int_0^\infty\int_{-\infty}^{-y}f(x)\mathrm dx\mathrm dy\Rightarrow\int_{-\infty}^0\int_0^{-x}f(x)\mathrm dy\mathrm dx$

Why does $$\int_{0}^{\infty} \int_{-\infty}^{-y} f(x)\mathrm dx \mathrm dy \Rightarrow \int_{-\infty}^{0} \int_{0}^{-x} f(x) \mathrm dy \mathrm dx$$ The title is pretty self explanatory. I couldn't ...
28 views

### For which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges?

Im looking for which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges. What I was looking for is an appropriate change ...
34 views

33 views

### Using change of variables to transform density functions

I'm was working on some exercises on statistical inference and came across a question I could not solve. After a while I decided to take a look at the solution to hopefully understand the problem ...
65 views

### $\int_{x^2+y^2+z^2 \leq 1}\frac{dx\,dy\,dz}{x^2+y^2+(z-2)^2}$

I'm trying to calculate the integral $$\int_{x^2+y^2+z^2 \leq 1}\frac{dx\,dy\,dz}{x^2+y^2+(z-2)^2}.$$ I've tried in two methods: Regular spherical coordinates, but this leads to really unfun ...
28 views

### Change of variable in Definite Integrals : Shift by value of Integral [closed]

For an integral of the form(given below) which does not have an anti-derivative $I = \int_{t=0}^\infty f(t)dt$ I wish to bound the value of the integral, by using a bound given in the problem ...
84 views

### How can this multiple integral be evaluated?

I am stuck trying to solve the following integral: $$\int_R (y+2x^2)(y-x^2) dA$$ where $R$ is defined by the following equations: $xy=1$, $xy=2$, $y=x^2$, $y=x^2-1$ with $x$ and $y$ positives. I've ...
If I have an integral of the form $$I = \int_{-T}^T dx\int_{-T}^T dy\ i(x - y) \tag1$$ Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I ...
### Use the spherical coordinates to compute the integral $\int\limits_{B} z^2 dx dy dz$ where B is defined by $1\leq x^2 + y^2 + z^2 \leq 4$
, however the answer I got to is different than the answer sheet. The answer sheet says that it should be $\frac{62}{15}$ Am I making some mistake or is the answer sheet incorrect?