0
votes
0answers
7 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
1
vote
0answers
17 views

Equivalent form of a double integral.

I am looking at the second question of this problem set: The iterated integral $\int_0^1 \int_{y/2}^1 e^{x^2} dx \, dy$ can be expressed as (a) $\int_0^1 \int_0^{2x} e^{x^2} dy \, dx$ ...
0
votes
1answer
31 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
0
votes
0answers
33 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
1
vote
2answers
39 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
0
votes
0answers
16 views

Integration by parts partial derivatives

Given $$\int_x \int_t \Big( \frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}f(u(x,t)) \Big) \phi(x,t)~~ dt dx = 0$$ How can I apply integration by parts in order to have the ...
1
vote
2answers
48 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty ...
2
votes
1answer
44 views

Compute a multiple integral$\iint_{[0,1]^2} (xy)^{xy} dxdy$

$$\text{Compute} :\iint_{[0,1]^2} (xy)^{xy} dxdy$$ I am thinking about changing the variable, $x=u,y={v \over u}$.But it doesn't work. I just found that the answer is$\int_0^1 t^t dt$.Maybe my idea ...
1
vote
0answers
35 views

How do I do this double integral (change of variable)

$B$ is the region bounded by $xy = 1$, $xy = 3$, $x^2 - y^2 = 1$, $x^2 - y^2 = 4$ Find $$\iint\limits_{B}x^2 + y^2 \,dx\,dy$$ using the change of variables: $$u = x^2 - y^2$$ $$v = xy$$ So I think ...
0
votes
1answer
68 views

Triple integration and transformations [on hold]

Hello I am a 17 year old kid in high school that does math for fun. One of my buddies gave me these problems and I can't seem to figure it out. Question 1 Find the volume of the finite region in ...
2
votes
2answers
31 views

How to prove this multivariable integral identity?

By numerical experimentation I found that $$ \lim_{\beta \rightarrow \infty} \frac 1 \beta \int_0^{\beta}dx \int_0^{\beta}dy \, f\left( |x-y| \right) = 2\int_0^{\infty}dx \, f(x) $$ if $f:\mathbb{R} ...
0
votes
0answers
15 views

Changing a double integral to single integral

I have seen this integration problem in a random process text book. We have the following integral. $\int_{-T}^{T}\int_{-T}^{T}C(t_1-t_2)dt_1dt_2 = \int_{-2T}^{2T}(2T-|\tau|)C(\tau)d\tau$ where ...
0
votes
0answers
22 views

Integration with cylindrical coordinates

I need to use cylindrical coordinates to find the volume of a region which when projected onto the xy plane is the disk $x^2+y^2+2y-3=0$. I already know what I'm integrating between for $z$ but I need ...
5
votes
1answer
30 views

Multiple Integration order doesn't agree.

Let $0<x,y,t,z<1$ with the additional condition: $$\begin{align*} x &< t\\ \wedge & \ \\ y &<z \end{align*}$$ Call the set of all $x,y,t,z$ satisfying the above conditions ...
1
vote
1answer
27 views

Stokes theorem problem $\displaystyle \int_C (3y+z)dx+(x^2 +2yz)dy+(2x+y^2)dz$

Let $ S_1=\{(x,y,z) \ | \ x^2+y^2-2x-2y+1=0 \} $ $ S_2=\{(x,y,z) \ | \ 2x+3y+z=9 \} $ and $C=S_1\cap S_2$ I'd like to calculate following integral $$ \int_C ...
1
vote
0answers
27 views

How to parametrize the volume of the intersection of cube and a right tetrahedron?

This is an extension of my previous question. I am trying to find the volume of the region which is the intersection of a cube given by $\vec r_1 = (x,y,z)$, where $$\begin{cases}0 \le x \le 1 \\ 0 ...
4
votes
1answer
39 views

Why can I not combine integrals this way?

Evaluating the triple integral $\int^1_0 \int^{1-z}_0 \int^{1-y-z}_0 \text{dxdydz}$, I get $\frac 16$. Evaluating the triple integral $\int^1_0 \int^1_0 \int^1_0 \text{dxdydz}$, I get $1$. So I ...
0
votes
2answers
34 views

Green's theorem exercise

I am trying to solve the following problem: Show functions $P,Q:\mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$ of class $C^1$ that verify $P_y=Q_x$ but $$\int_\gamma P(x,y)dy+Q(x,y)dy \neq 0$$ where ...
0
votes
1answer
32 views

Why i got negative value for volume?

I want to find the indicated volumes under the surface $z=\frac{1}{y+2}$ and over the area bonded by $y=x$ and $y^2+x=2$. After sketching the graph for $x=2-y^2$, and $x=y$ i found that $y=0$ and ...
1
vote
1answer
33 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
1
vote
2answers
46 views

How to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis?

Given $\int \int dxdy$, I want to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis. I think the limits of integral in $y$ axis are from $y=\ln\left(x\right)$ to ...
3
votes
0answers
12 views

Multiple Integral Substitution Error

I just started learning about the substitution rule for multiple integrals and I decided to give myself an example problem: Calculate $\iint_R{(x^2 + y^2)dA}$ with $R = \{(x, y) \in \Bbb{R} \ |\ 0 ...
1
vote
1answer
16 views

Double integral in cylindrical coordinates

I'm having trouble with a double integral problem in cylindrical coordinates. I'm sure the answer is staring me in the face, but I'm missing something. In the multivariable version of the Community ...
1
vote
1answer
25 views

what is the problem with this variable transformation?

$$\iint\limits_D (x − y)^2 \sin^2(x + y) \, dx \, dy$$ where $D$ is a parallelogram with vertices at $(π, 0), (2π, π), (π, 2π)$ and $(0, π)$. We can change the variables as $$x=\frac{u-v}{2} \text{ ...
1
vote
1answer
24 views

How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$ How may I describe E in cylindrical and spherical coordinates? And how may I ...
0
votes
0answers
10 views

Model and compute the area A of the region inside of C

Let a > 0. Let C denote the set of points P in the plane such that the distance from P to (-a, 0) times the distance from P to (a, 0) is a^2. The set C is a curve. Model and compute the area A of the ...
0
votes
1answer
33 views

Having trouble proving this integral is infinite

I am working on an assignment and I have to prove that $$\frac{x^2-y^2}{(x^2+y^2)^2} \notin L_1\mathbb{R}^2)$$ to justify why Fubini's Theorem does not apply. I figured that the best way to do this ...
2
votes
0answers
28 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between ...
0
votes
0answers
40 views

The partial derivatives of the function $s=\int _u^v\frac{(1-e^t)}{t}dt$

If $s=\int _u^v\frac{\left(1-e^t\right)}{t}dt$, I want to find $\frac{∂s}{∂v}$ and $\frac{∂s}{∂u}$ and their limits as $u$ and $v$ tend to zero. first i find for $\frac{∂s}{∂v}=\frac{∂}{∂v}\int ...
1
vote
1answer
42 views

Green-Riemann Theorem

I calculated the circulation of the vector field : $$\vec{v} = -y\omega \, \vec{i} + x\omega \, \vec{j}$$ over the ellipse : $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ I found $2 \pi \omega a b$. ...
1
vote
0answers
30 views

Gauss /divergence theorem: generalization

The divergence theorem / Gauss integral theorem states that $\int dV\; \nabla \cdot \vec F = \int dS\; \hat n \cdot \vec F$ for a vector function $\vec F$, with $dV$ the volume element, $dS$ the ...
0
votes
2answers
57 views

Finding volumes with double integrals

I got some troubles with this problem: A swimming pool is circular with $20\,ft$ diameter. The depth is constant along east-west lines and increases linearly from $4\,ft$ at the south end to $9\,ft$ ...
0
votes
0answers
16 views

gradient of an integral with variable in the upper limit

Is the value of the following computation true? Let $x,y,z\in R^{2n}$ and the continuous vector-valued function $F:\mathbb{R^{2n}}\to\mathbb{R^{2n}}$. Then $\nabla_y \left(\int_{0}^{y}F(x,z)\cdot ...
1
vote
0answers
19 views

Solve $\int\limits_{\mathbb{R}^n} e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$

How to calculate $$\int_{\mathbb{R}^n}e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$$ where $\langle \cdot, \cdot \rangle$ denotes the canonical inner product in $\mathbb{R}^n$. I'm trying use ...
0
votes
1answer
63 views

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ prove the following:

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ such that $ f(0)\neq -1$ and $\displaystyle\int_{0}^{b} f(t) \, dt=0$ Show that the equation $\displaystyle\int_{x}^{a} f(t) ...
2
votes
2answers
67 views

help with strange Double Integral: $\iint_E {x\sin(y) \over y}\ \rm{dx\ dy}$

i'm having trouble with this double integral: $$ \iint_E {x\sin(y) \over y}\ \rm{dx\ dy},\ \ \ \ E=\Big\{(x,y) \in \mathbb{R^2} \mid 0<y\le x\ \ \ \land\ \ \ x^2+y^2 \le \pi y\Big\} $$ i've ...
1
vote
1answer
49 views

A version of the Fundamental Theorem of Calculus for two variables

Let $f(x,y)$ be differentiable in the rectangle $R=[a,b]\times[c,d]$, show that the function $\displaystyle F(x,y)=\int_{a}^{x} f(t,y) \, dt$ is also differentiable in $R$ and that ...
2
votes
3answers
63 views

Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta $ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
1
vote
1answer
27 views

How to set up an integral by these conditions?

I've got these surfaces: $$ z = 0\\ z = 4 - y^2 $$ And a cylinder: $$ x^2+y^2=4 $$ I need to find the volume enclosed by these figures. As far as I understand the limits of integration for $z$ are ...
2
votes
1answer
52 views

A proof involving nested integrals and induction [duplicate]

Prove that $$\int_0^x dx_1 \int_0^{x_1}dx_2 \cdots \int_0^{x_{n-1}}f(x_n) \, dx_n =\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t) \, dt$$ I'm trying induction over $n$. The case $n=1$ is trivial. When ...
0
votes
1answer
29 views

Set up triple integral for volume (cylindrical coordinates)

I am given the following question Let $D$ be the region in $\mathbb{R}^3$ that lies within $x^2 + y^2 =4$, underneath the surface $z= 4- x^2 - y^2$ and above the surface $z=- \sqrt{9-x^2 - y^2}$ ...
1
vote
2answers
37 views

Finding the Limits of the Triple Integral (Spherical Coordinates)

Let $D$ be the region in $\mathbb{R}^3$ below $z=-\sqrt{x^2 + y^2}$ and above $z=-\sqrt{4-x^2 -y^2}$. Rewrite \begin{align*}\iiint \limits_D z^2 dV\end{align*} using Spherical Coordinates. I ...
1
vote
1answer
33 views

Volume of a solid bounded by surfaces - is it correct?

Could you check if my calculations and reasoning are correct. And maybe suggest a nicer way of solving this problem? We are given a solid bounded by these surfaces: $y=x^2, \ y=1, \ 2x+y+z = 4, \ ...
1
vote
2answers
50 views

Double integral of $\dfrac{y}{x^2y^2+1}dx~dy$

I'm trying to solve the double integral $\displaystyle\int_0^1\int_0^1\dfrac{y}{x^2y^2+1}dx~dy$ . I'm guessing something with natural log will have to be done. Doing the steps of this problem are more ...
3
votes
2answers
120 views

Proof of Gauss theorem (divergence theorem) in $\mathbb R^2$

I am trying to solve an exercise in where it is asked to show the divergence theorem, or also known as Gauss theorem, in $\mathbb R^2$ using Green's theorem. I suppose that the divergence theorem in ...
0
votes
1answer
37 views

Bounding the average of a vector valued function

Disclaimer: I edited the question so that it fits Daniel Fischer's comments and it becomes more general. I also provide an answer myself in case anyone might be interested in the solution. Question: ...
1
vote
3answers
56 views

Calculating volume enclosed using triple integral

Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$ I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of ...
4
votes
1answer
99 views

Area and volume relation (multivariable calculus problem)

Let $D \subset R^3$ a region over the plane $z=0$, if $C$ is the cone of base $D$ and vertex at $(0,0,1)$, show that $Vol(C)=\dfrac{1}{3}A(D)$, where $A(D)$ is the area of the region $D$. First I ...
0
votes
1answer
23 views

A doubt about line integral as a manifold.

I'm trying do a conection between a definition that I've learned in my graduation and nowadays a definition that I've learned in my doctorate studies. Calculus Definition: Let $C$ be a curve ...
0
votes
1answer
36 views

Closed simple curve and curl of function

Let $F: \mathbb R^3\setminus \{0\} \to \mathbb R^3$. Prove that if $\operatorname{curl}(F)=0$, then the integral $\int_C F\cdot ds=0$ for every simple closed curve $C$. Is this statement true for ...