12
votes
0answers
108 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
1
vote
2answers
67 views

Triple Integral exercise

Calculate $\int\int\int_Dz\;dxdydz$ if $D$ is the region inside $z=0,z=\sqrt{x^2+y^2}$ and $x^2+y^2=1$. I would like to know if the answer I got is right. This is what I did: $(1)$ Change to ...
1
vote
1answer
71 views
2
votes
3answers
147 views

Trouble computing this double integral

$$\iint_R xe^{xy}~\mathrm{d}A \qquad 0\le x\le 2 \quad 0 \le y \le 1$$ Today I started learning about double integrals on a class I am taking, had good understanding on single-variable integrals but ...
0
votes
1answer
27 views

exchanging partial derivative and an integral

Does $\frac{\partial }{\partial x}\int_{u(x)}^{v(x)}f(x,t)dt$ = $\int_{u(x)}^{v(x)}\frac{\partial }{\partial x}f(x,t)dt$ ?? If yes, how did we do that although there are 2 functions $u$ and $v$ ...
1
vote
0answers
19 views

Integrals depending upon a parameter

There was an exercise, in my professor's book, asking to prove the continuity of an integral depending upon a parameter. Namely, the hypothesis were: Let $D$ be a measurable subset of $\mathbb{R}^n$, ...
1
vote
1answer
48 views
0
votes
3answers
62 views

Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
0
votes
2answers
38 views

Double Integration: $\iint_D\ e^{30x}\ dA$

I am having trouble with this double integral. I know I must set it up to have the $y$ values go from $x$ to $x+1$ and the $x$ values from $0$ to $1$. When I solved the integral I got the answer ...
0
votes
1answer
30 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
0
votes
2answers
42 views

Integrating $\iint_R \sin(9x^2+4y^2)\ dA$

The question I'm trying to solve is: $\displaystyle \iint_R \sin(9x^2+4y^2)dA$, where $R$ is the region in the first quadrant bounded by $9x^2+4y^2=1$. I'm a little confused in solving this. Does ...
1
vote
0answers
40 views

Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...
0
votes
1answer
40 views

Use divergence theorem to find $\iint_S (2x+2y+z^2) dS$ Where $S$ is the sphere $ x^2+y^2+z^2 = 1$

I tried a lot but it gets ugly really soon, any help will be greatly appreciated. T hanks
3
votes
1answer
59 views

Interesting dilemma, answer not matching with stewart, My work is Included

Question : Compute flux through the upper hemisphere of $x^2+y^2+z^2 = 1$ . Where $$\textbf{F} = \left( z^2x\right)\textbf{ i }+\left[\dfrac{1}{3}y^3+ \tan z\right]\textbf{ j } + \left(x^2z+y^2 ...
0
votes
3answers
41 views

Triple integral calculation gone wrong?

Calculate the volume of the solid bounded $K$ by $$z \geqslant x^2 + y^2 - 1, \quad z \leqslant \sqrt{x^2+y^2} + 1, \quad z \geqslant 0$$ The triple integral I set up is $$ \iiint_K ...
2
votes
1answer
44 views

Calculate $\int_D x^3y\ dx\,dy$

Let $D$ the bounded region by the $y$-axis and the parable $x= -4y^2 + 3$. How can I calculate the integral $$\int_D x^3y\ dx\,dy$$ I am stuck with this problem some help to solve this please.
1
vote
2answers
41 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
2
votes
1answer
45 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
0
votes
1answer
53 views

Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Prove without using the divergence theorem. The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem. Thanks in advance.
3
votes
3answers
79 views

calculate $\int_{0}^{\pi} \int_{0}^{x}\log(\sin(x-y))dydx$

I was asked to find the integral $\iint_A \log(\sin(x-y))dxdy$ where $A$ is the triangle $y=0, x=\pi, y=x$ in the first quadrant. I was given a hint: evaluate $\int_{0}^{\pi}\log(\sin(t))dt$ using ...
2
votes
4answers
62 views

Mass of elipsoid

We are given the elipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \leq 1$ with density function $p(x,y,z)=x^2+y^2+z^2$. Find the mass of the elipsoid. what I did: I used the ...
0
votes
2answers
39 views

how do we interpret this integral from polar co-ordinates

$$\text{Find } \int_C rdr$$ Where $C$ is any closed loop. I feel that the answer is zero, i have no hard reasoning. Here $r$ is the parameter from the polar coordinates.
1
vote
3answers
102 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
1
vote
1answer
35 views

Integrate $f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$

$f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$ I have to integrate it over $E=[0,1]\times[0,1]$ I can integrate the inner integral ...
1
vote
0answers
37 views

Imaginary part of a triple intragral

I'm only an introductory calculus student(I just finished my high school AP calculus BC course) so I apologize if these questions seem a bit elementary. I'm using Mathematica to perform a triple ...
0
votes
2answers
40 views

How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
2
votes
1answer
35 views

Stokes theorem: $ \vec{v} = (xz, -y, x^2y)$

Evaluate $$ \iint_S \nabla \times \vec{v} . \vec{N} dS$$ where $\vec{v} = (xz, -y, x^2y)$ and S consists of the 5 faces, not on xy plane, of the cube$ [0,2] \times [0,2] \times [0,2] $ and $ ...
0
votes
0answers
12 views

Gauss theorem: $\vec{v} = (x^2 + ye^z, y^2 + ze^x , z^2 + x e^y)$

If D is the region bounded by the cylinder $x^2 + y^2 = 1, z=0$ and $z= x+2$, use the Gauss Theorem to evaluate $$ \iint_S \vec{v} . \vec{n} \; dS$$ where $S$ is $\partial D$ , $\vec{n} $ points to ...
0
votes
3answers
35 views

Surface Integral of $\frac{\vec{r}}{|\vec{r}|^3}$

Let $$\vec{v} = \frac{\vec{r}}{|\vec{r}|^3},$$ $\vec{r}=(x,y,z)$. Evaluate $$ \iint\limits_S \vec{v} \cdot \vec{n} \, dS$$ with $\vec{n} $ pointing to the exterior of the surface $S$ ...
0
votes
1answer
26 views

Surface integral over a rectangle

Evaluate $$ \iint_S \vec{v} . \vec{n} dS$$ for $\vec{v} = (x+y, -2y - 1, z)$ and $S$ the rectangle of vertices $(1,0,1),(1,0,0),(0,1,0),(0,1,1)$ and $\vec{n}$ points in the opposite direction of ...
0
votes
1answer
23 views

Surface integral over the plane $x+y+z=2$

Evaluate $$\iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy$$ where $S$ is the part of the plan $x+y+z=2$ in the first octant, with normal $n$ such that $n . (0,1,0) \geq 0$ My ...
0
votes
1answer
32 views

Surface integral over a sphere - parametrization

Evaluate the surface integral of the field $A(x,y,z)=(xy, yz, x^2)$ over the sphere $S$ givn by $x^2 + y^2 + z^2$ with the normal vector pointing to the exterior of the sphere I've tried doing this ...
3
votes
1answer
61 views

approximate this fancy looking double integral

$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
2
votes
1answer
87 views

This double integral

$$ \int_0^1\int_0^1x^3y^2\sqrt{1+x^2+y^2}\hspace{1mm}dxdy$$ We have to compute this up to 4 decimal places
2
votes
0answers
20 views

help finding this surface integral

Given that $$S : z = xe^y, \hspace{1mm} 0 \leq x\leq 1, \hspace{1mm} 0 \leq y\leq 1 $$ Find $$\int\int_S \left(x^2+y^2+z^2\right)dS$$ upto four decimal places
1
vote
1answer
38 views

Cylindrical triple-integral bounds

The integral to solve is as follows: $$\iiint(x^2-y^2)^2 dV$$ Bounded in the first octan by $z=\frac{1}{x^2+y^2},x+y=2,x+y=1$. In order to solve it and practice transition to cylindrical ...
4
votes
4answers
80 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
0
votes
1answer
35 views

Volumes of solids of revolution - guide to solve similar problems

So, I'm preparing for an exam and I'm stuck with problems with volumes of solids of revolution. I have two examples: a, find the volume of a solid, described as: $T=\{(x,y,z) : x^2 + y^2 - z^2 ...
2
votes
3answers
92 views

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found?

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy $ be found, if $D$ is $x$ O $y$ axis? So far I have done it this far: ...
0
votes
0answers
30 views

Integral of multivariate normal density function

Is anybody know a suited close-form solution for this integral: $$ I=\int_{R^n} x_i \cdot x_j \cdot f_N({\bf x},{\bf \mu},{\bf \Sigma}) d{\bf x} $$ where ${\bf x}=\{x_1,\ldots,x_n\}$ and $f_N$ is the ...
1
vote
2answers
58 views

Procedure for evaluating $\int_{x=\ -1}^1\int_{y=\ -\sqrt{1-x^2}}^{\sqrt{1-x^2}}\frac{x^2+y^2}{\sqrt{{1-x^2-y^2}}}\,dy\,dx$

While solving another problem I have come across this integral which I am unable to evaluate. Can someone please evaluate the following integral? Thank you. $$\int_{x=\ -1}^1\int_{\large y=\ ...
1
vote
1answer
97 views

Double Integral $\iint_D\ (x+2y)\ dxdy$

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like? You mean something like this? ( I made mess) $$\iint_D ...
1
vote
1answer
50 views

Evaluate double integral:

Evaluate: $$\iint_{D} \arcsin(x^2+y^2)\, dx\,dy$$ where $D$ is defined by the following polar equation $\rho=\sqrt{\sin \theta}$ and $0\le\theta\le\pi$
12
votes
10answers
404 views

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
1
vote
2answers
61 views

How to integrate $e^{-z}$ over the ball $x^2 + y^2 + z^2 \leq 1$

I can't figure out how to obtain the limits for this triple integral since I can't visualize how $e^{-z}$ varies over a sphere in my head.
1
vote
1answer
34 views

Definition of an integral over a domain.

In calculus we generally use this notion: $\int_D f(x)\,dx$. I understand that when $D$ is an interval from $a$ to $b$ the integral is equivalent to $\lim \limits_{\|\Delta x\| \to 0} \sum_{i} f(x_i) ...
2
votes
1answer
69 views

How to integrate $\int_{0}^{1} \int_{0}^{\pi} \int_{0}^{\pi} r^2 \sin\theta \sqrt{1 - r^2\cos^2\theta - r^2\sin^2\theta} \,d\phi\, d\theta \,dr$

Find the center of mass of the hemispherical region $W$ defined by the inequalities $x^2 + y^2 + z^2 \leq 1$ and $z \geq 0$ with unity density. By symmetry we know the $x$ and $y$ coordinates ...
2
votes
1answer
63 views

Line integral: $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$

Let $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$ and D the domain bounded by the torus obtained by rotating the circunference $(x-2)^2 + z^2 =1, y=0$ around the z-axis. Show that $rot( u )=0 ...
1
vote
4answers
88 views

Double integral for $\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$

I'm trying to evaluate this $$\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$$ tried substition $$ u = {(x^2+y^2+1)}^{-1} \ \ du = \ln {(x^2+y^2+1)}$$ but du is not found in the ...
0
votes
2answers
36 views

How to find the limits of a triple integral converted to spherical coordinates

Find the integration limits of $\int_{0}^{3} \int_{0}^{\sqrt{9 - x^2}} \int_{0}^{\sqrt{9 - x^2 - y^2}} \frac{\sqrt{x^2 + y^2 + z^2}}{1 + (x^2 + y^2 + z^2)^2} dz dy dx$ in spherical coordinates. ...