All Questions
Tagged with bounds-of-integration multiple-integral
21
questions
0
votes
1
answer
77
views
how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
1
vote
1
answer
31
views
Probability densities with conditions - how to find the distribution function
I have two probability density functions where i need to find the distribution function.
The first function is
$$f(x,y)=
\begin{cases}
\frac{x}{y} & \text{for $0\leq x\leq y\leq c$}\\
0&\text{...
- 11
0
votes
0
answers
30
views
Triple Integral - Use symmetry for center of mass question?
I am unsure when to use symmetry with triple integrals.
Can I use symmetry for this centre of mass question?
$E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; ...
- 143
0
votes
1
answer
41
views
Triple integral (mass) - setting up region between planes and parabolic cylinder
I am trying to set up the following triple integral using the xy plane.
$E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+6z=6, x=0$, and $z=0 ; \quad \rho(x, y, z)=8$.
I set up ...
- 143
0
votes
1
answer
92
views
How does one calculate the area of a set?
The set is $M=\{(x,y)\in\mathbb{R}^2:|x|+|y|\leq 1\}$.
Question: How do you calculate the area of $M$? More specific, how do you find the bounds of integration?
Attempt: I tried to solve the ...
- 1
1
vote
1
answer
115
views
Evaluation of $~\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y~$ where $~D~$is of bounded and closed and line symmetric with $~y=x$
$~D:=$domain where it is bounded and closed in $~ \mathbb R^2 ~$ and line symmetric with $~ y=x ~$
$$
I:=\underbrace{\iint_{D} {x^2-y^2 \over 1+x^4+y^4 } \mathrm{d}x \mathrm{d}y}_{\text{I want to ...
- 1,519
2
votes
2
answers
113
views
Double integral of $1/(x^2+y^2)$ restricted to $x^2+y^2\leq2$ and $x\leq1$
Find $$\iint_D \frac{1}{(x^2+y^2)^2}dA$$ where $$D = \left\{ (x,y): x^2 + y^2 \leq 2 \right\} \cap \left\{ (x,y): x \geq 1 \right\}$$
Because of the prevalence of $x^2+y^2$ terms here, I figured we ...
- 23
1
vote
0
answers
44
views
Integral bounds for $x\geq yz$
I am having trouble understanding the integral bounds.
From what side should my understanding go (first or second?):
first: as $z$ is between zero and one, $y$ is also between zero and one, thus $x$ ...
- 2,527
0
votes
2
answers
419
views
Double integral setup - Uniform distribution
I am currently trying to understand a specific component of a probability problem involving setting up the proper bounds on a double integral.
In the problem, $X_1$ and $X_2$ are independent Uniform $(...
3
votes
3
answers
889
views
$\iiint_M (x+y+z)\,dx\,dy\,dz$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$
$$\iiint_M (x+y+z)\,dx\,dy\,dz$$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$. How would I express this with the correct bounds? Once I have the bounds I can continue on my own but I need ...
- 487
0
votes
2
answers
634
views
Evaluate $\iiint_{R} (2x+y) \,dx \,dy \,dz$
Evaluate $$\iiint _{R} (2x+y) \,dx \,dy \,dz\,,$$ where $R$ is the region bounded by the cylinder $z = 4 - x^{2} $ and the planes $x = 0$, $y = 0$, $y = 2$ and $z = 0$.
How do I extract the limits ...
- 7,426
3
votes
1
answer
6k
views
Evaluating $\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$
Question: Evaluate the given triple integral with cylindrical coordinates:
$$\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$$
My solution (attempt): Upon ...
user776254
0
votes
1
answer
131
views
How to calculate double integral that involve the max function?
I would really appreciate if you could help me to solve this integral or at least determine the bound since I do not know if it converge or not of the integral:
$\iint\limits_R {Exp\left[ { - \max \...
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