Questions tagged [bounds-of-integration]
In many questions the problem of determining bounds of integration in multiple integrals is a major part of what an answer needs to deal with, and in surprisingly many questions it is the only issue. This tag is for such occasions.
123
questions
0
votes
2
answers
65
views
Sum of two independent random variables: Inconsistent CDF
I have two random continuous RV $X$ and $Y$ and the sum $Z=X+Y$. I am trying to derive the pdf from the first principle but somehow I could not get the results to agree.
--
Let $Y\sim\mathcal{U}(0,1)$ ...
- 4,333
1
vote
1
answer
52
views
Finding $\iint_D \frac{10}{ \sqrt {x^2 + y^2}}dx\,dy$ for $D = \{(x,y) \mid 0 \leq x \leq 1, \sqrt{1-x^2} \leq y \leq x\}$
As I said title,
$$\iint_D \frac{10}{ \sqrt {x^2 + y^2}}\,dx\,dy$$ for $$D = \{(x,y) \mid 0 \leq x \leq 1, \sqrt{1-x^2} \leq y \leq x\}$$
I tried it using integration by substitution by $(x,y) = (r\...
- 2,246
2
votes
0
answers
69
views
Help with change of variables for evaluating $\iint_S (x^2-y^2)e^{(x+y)^2} \,dx\,dy$
I have $$\iint_S (x^2-y^2)e^{(x+y)^2}\,dx\,dy $$ with restrictions $x+y\leq3$, $ xy\geq2$ and $y\leq x$
I think that with the variable changes $$u=x+y$$ and $$v=x-y$$
whose Jacobian is $2$
then I ...
- 31
2
votes
1
answer
115
views
How to find the bounds of the volume integral $\int_\Omega (6xz + 2y +3z^2) \, \text{d} \tilde{x}$?
I'm studying on integrating over volumes and I don't know how to set the bounds in this exercise:
Let $\Omega := \left\{ (x,y,z) \in \mathbb{R}^3 \,\big| \,\frac{x^2}{4} + y^2 + \frac{z^2}{9} <1 \...
- 195
1
vote
1
answer
386
views
What does this integral notation mean?
I'm talking about the integral part that is highlighted:
Should I interpret the top one highlighted as the upper bound of integration and the bottom one as the lower bound? That's the only ...
- 195
1
vote
1
answer
153
views
A lower bound for the cosine integral
I am reading Devroye's paper 2001Simulating Perpetuities. On P103, he mentioned a lower bound for cosine integral, i.e.,
$$\int_0^t\frac{1-\cos s}{s}ds \geq max(0,\gamma+\log t),$$
where $\gamma$ is 0....
- 887
2
votes
3
answers
99
views
How to evaluate $\iint_R \sin(\frac{y-x}{y+x})dydx$ with Jacobian substitution?
I want to calculate this integral with substitution $x=u+v , \ y=u-v$:
$$\iint_R \sin\left(\frac{y-x}{y+x}\right)dydx$$
$$R:= \{(x,y):x+y≤\pi, y≥0,x≥0\}$$
but I don't know how to set new bounds for $...
- 301
0
votes
2
answers
149
views
Does the existence of the integral $\int_0^\infty f(x)dx$ imply that f(x) is bounded on $[0,\infty)$ when f(x) is continuous in this same interval?
Does the existence of the integral $\int_0^\infty f(x)dx$ imply that f(x) is bounded on $[0,\infty)$ when f(x) is continuous in this same interval ?
edit
I'm trying to use Cauchy without knowing ...
- 29
1
vote
1
answer
197
views
Limit of ratio of incomplete gamma function
In order to derive Sterling's approximation, I need to show that the following integral decays quicker than at least $\mathcal{O}(n^2)$:
$\lim_{n\to\infty}\dfrac{\int_{2n}^\infty x^ne^{-x}dx}{\int_{0}^...
- 185
0
votes
1
answer
40
views
$D(x,y)=\frac{xy}{x^3+y^3}$ verifies $\int_0^1 D(x,y) dx \leq c$
Let $D(x,y)=\frac{xy}{x^3+y^3}$. In order to prove that the operator
$$T_D : L^2(0,1) \to L^2(0,1), \quad f \mapsto T_Df(x)=\int_0^1 D(x,y)f(y)dy
$$
is bounded, I need to show that there exists some $...
- 469
2
votes
1
answer
1k
views
Choosing Bounds of Integration for a Triangle
Suppose I have a region of integration boded by $x+y \le 8$ and $0 \le y \le x$. I have graphed the bounds, and they form a right triangle with its hypotenuse following the $x$-axis from $[0,8]$ and ...
- 657
0
votes
1
answer
62
views
Changing the order of integration: How to set the bounds
A derivation is given in a textbook. Here, $F$ is a probabilistic distribution function, which means that: $$\int_{-\infty}^{\infty} F(t) dt = 1$$
At one step, they change the order of integration:
\...
- 369
0
votes
1
answer
255
views
Convolution of function... integral bounds?
Okay so for this question: Convolution of a function with itself
The answer stated that in the case of $x\le 0$:
the integral bounds are from 0 to x. Why is this?
I also don't understand why from $...
- 41
11
votes
3
answers
538
views
Is it necessary to write limits for a substituted integral?
To solve the following integral, one can use u-substitution:
$$\int_2^3 \frac{9}{\sqrt[4]{x-2}} \,dx,$$
With $u = \sqrt[4]{x-2}$, our bounds become 0 and 1 respectively. Thus, we end up with the ...
- 121
0
votes
0
answers
27
views
Bounds of integration for $E[\frac{X}{Y}]$. $X,Y$ are exponential r.v. with parameter alpha and beta.
Let $X,Y$ be Exponential random variables with parameter alpha=lambda and beta=mu. Assume $t=u$.
Appendix: Evaluation of the integral:
\begin{align}
f_{X/Y}(u) = {} & \alpha\beta \int_0^\...
user739402