2
votes
1answer
39 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
0
votes
0answers
23 views

find the volume of the region using triple integrals using cylindrical coordinates

The volume of the pyramid defined by $(0,0,0)$, $(2,0,0)$, $(0,1,0)$ and $(0,0,4)$. Calculate: $\displaystyle\iiint(2+z^2)\,dV$ The limit of the radius is where I am stumped, since the $xy$ ...
1
vote
1answer
25 views

Triple integral question

In a textbook problem, I am asked to find the flux through a given surface using the divergence theorem. That is, $$\iiint_{G} \nabla \cdot \vec{F} \, dV = \iint_{\sigma} \vec{F} \cdot \vec{n} \, dS$$ ...
1
vote
3answers
72 views

Is the definite integral of the function necessarily the anti-derivative?

Let's say you have a function defined as $$g(x)=\int_1^xf(t)dt$$ By the integral definition, g(x) is the area under the curve of f(x) from 1 to x. eg: g(5) is the area under f(x) from 1 to 5. I ...
1
vote
1answer
35 views

Double Integral with abstract functions of 2 variables

I am required to prove something, and so far I have come to set up an integral $$\int_0^l{\int_0^T{u(x,t)\, \frac{d}{dt}u(x,t) dt }dx}.$$ I was just wondering how to think about these ...
1
vote
2answers
47 views

Evaluating $\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA$

I'm trying to evaluate the double integral $$ \frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{\frac{-x^2}{2}} e^{\frac{-y^2}{2}} \, dA. $$ Any ideas?
0
votes
1answer
28 views

Surface integral question help.

If $S$ is the surface of the sphere $x^2 + y^2 + z^2 = a^2$, compute the value of the surface integral $$\iint_S xz\,{\rm d}y\,{\rm d}z + yz\,{\rm d}z\,{\rm d}x + x^2\,{\rm d}x\,{\rm d}y$$ ...
0
votes
2answers
21 views

Integrate over triangle

Integrate $f(x,y) = (x+y+1)^{-2}$ over the triangle with vertices $(0,0), (4,0), (0,8)$. I think you have to split up the triangle into different equations: $x=0, y=0, y=2x-8$. But I'm not sure what ...
0
votes
2answers
31 views

Double Integral - Sketch region and evaluate

Sketch the region of integration and evaluate the integral: $$\int_1^2 \int_y^{y^2} dx \, dy$$ I understand how to take the integral, but the region of integration seems like it has no bounds. Like ...
1
vote
2answers
46 views

How could I solve this double integral question

Evaluate$$ \iint_R \left ( e^{-x-y} \right )dxdy, $$ where $R$ is the region in the first quadrant in which $x+y\leq 1$. I think the first step is $$\int_{0}^{1}\int_{0}^{1-y}\left ( e^{-x-y} ...
2
votes
1answer
59 views

Find the volume of the solid bounded by the surface given in spherical coordinates by $R = 4-3\cos(\phi)$.

It is worth noting that $R$ in this case denotes the distance from origin to a point $P$ in space. You may be more familiar with $\rho$ instead of $R$. Here is my attempted solution: I am assuming ...
2
votes
1answer
38 views

SOLVED: Green's theorem result and line integral result are not equal! What am I doing wrong?

I have this line integral: $\oint 3ydx+x^2dy$ and the path is a line from $(0, 0)$ to $(1, 0)$ (so this is $y=0$), another line from $(1, 0)$ to $(1, 1)$ (so this is $x=1$) and a curve $y=x^2$ from ...
2
votes
1answer
85 views

Flux and Gauss theorem

I have a problem; There seems to be something wrong with my understanding of gauss theorem. Let's say $F = [y ; x^2y; y^2z]$. I want to calculate the flux of $F$ going out of $$D = \{1 \le z \le 2 - ...
2
votes
0answers
37 views

How to simplify path integral?

I am trying to integrate a function, $f(x,y)$, over the straight line path connecting $(0,k)$ to $(k,0)$ in the x-y plane, where $k>0$ (the diagonal part of the boundary of a simplex in ...
1
vote
2answers
42 views

Finding work via Line Integrals

The position of an object with mass $m$ at time is $r(t) = at^2 \vec{i} + bt^3 \vec{j}$, where $0 \leq t \leq 1$. Part a asks for the force, which I found to be $2ma \vec{i} + 6mbt \vec{j}$, which is ...
2
votes
2answers
43 views

Triple integral over a sphere with parameter $2n$?

I need to intergrate $x^{2n}+y^{2n}+z^{2n}$ over a sphere of equation $x²+y²+z²=1$. I have thought of changing the coordinates from cartesian to spherical but I don't know how to deal with the ...
2
votes
3answers
94 views

Center of Mass via integration for ellipsoid

I need some help with the following calculation: I have to calculate the coordinates of the center of mass for the ellipsoid $$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( ...
0
votes
1answer
35 views

Is there a change of variables substitution that would allow this integral to be evaluated?

Is it known whether or not there exists a change in variable substitution that would allow the following integral to be evaluated? $$\int_0^1\int_0^1\int_0^1\frac{1}{1-xyz}dxdydz$$
2
votes
1answer
69 views

Integration limits of the double integral after conversion to the polar coordinates

I want to solve the following double integral: $$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$ And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and ...
2
votes
2answers
53 views

Area with double integral.

$f:\mathbb{R}^2\to\mathbb{R}, f(x,y)=(x^2+y^2)^2-8(x^2-y^2).$ Find the critical points of $f$ and the area of $X=\{(x,y):f(x,y)\leq 0\}$. To find the critical points I just have to find ...
1
vote
1answer
69 views

Multivariable Calculus Order of Integration Question

I have a triple integral $ \int_{0}^{2} \int_{0}^{y^3} \int_{0}^{y^2} f(x,y,z) \ dz \ dx \ dy $. I have to find five different iterated integrals equivalent to this integral. I know that order of ...
0
votes
0answers
26 views

what is the result of this Integral with polar coordinates?

I can't understand where am I going wrong with this integral. The final answer should be 4 but I get 2/3. Am I wrong or is the teacher wrong? Given this function: $f(x,y)=x+y$ and this domain D ...
1
vote
1answer
42 views

Consider the integral $I = \int_0^1 \int_{\sqrt{y}}^1 \frac{y(e^{x^2})}{x^3} dx dy$

Calculate the iterated integral by first reversing the order of integration $$\displaystyle I = \int_0^1 \int_{\sqrt{y}}^1 \frac{y(e^{x^2})}{x^3} dx dy$$
5
votes
1answer
118 views

Finding the volume inside an elyptical cylinder and a sphere

I'm trying to find the volume bounded by a sphere and an elyptical cylinder. The sphere is given by $x^2+y^2+z^2=1$ and the elyptical cylinder by $2x^2+y^2-2x=0$. My first attempt with spherical ...
2
votes
1answer
53 views

Triple integral with a cone as a domain

How can I find $\displaystyle\iiint_D f$ if $f(x,y,z) =\sqrt{x^2+y^2}$ $D$ is what is inside of $z^2=x^2+y^2,z=0,z=1$?. I tried to do it with cylindrical coordinates as follows: $x=\rho\cos\theta, ...
1
vote
1answer
37 views

Integration with cylindrical coordinates: Should I split the integral?

I have $f(x,y,z) = x^2+y^2$ and $D=\{(x,y,z): (x,y,z) \text{ are points inside } x^2+y^2=2x \text{ and between} z=0,z=2\}$ The equation $x^2+y^2=2x$ is equivalent to $(x-1)^2+y^2=1$ which ...
1
vote
0answers
50 views

An attempt to a triple integral with spherical coordinates.

If $f(x,y,z)=xyz$ and $D=\{(x,y,z)\in\mathbb{R}^3:x\geq 0, y \geq 0, z \geq 0,\; x^2+y^2+z^2\leq 1\}$, find $$ \iiint_D f(x,y,z). $$ I tried to solve this with spherical coordinates: ...
3
votes
2answers
78 views

Find the volume between two surfaces

Find the volume between $z=x^2$ and $z=4-x^2-y^2$ I made the plot and it looks like this: It seems that the projection over the $xy$-plane is an ellipse, because if $z=x^2$ and $z=4-x^2-y^2$ ...
2
votes
1answer
49 views

Finding a volume

Find the volume of $D\{(x,y,z)\in \mathbb{R}^3:\frac{x^2}{a^2} +\frac{y^2}{b^2}\leq z\leq 1 \}$ It looks like (1) I believe this could be solve with a double integral an considering the ...
2
votes
2answers
83 views

Double integral — tricky?

If $f(x,y) = x^2+y^2$ and $D=\{(x,y)\in\mathbb{R}^2:x^2+y^2\geq1, x^2+y^2-2x\leq0 \text{ and } y\geq0\}$, find $\displaystyle\int\displaystyle\int_D f$. $D$ looks like the intersection between ...
0
votes
3answers
53 views

A question about a simple integral.

How could I show that the $$\iint\sin(x)dxdy$$ along the domain $$x^2+y^2\leq1$$ is zero? I tried using polar coordinates but to no avail. Had thought about claiming Sine is an odd function so the ...
0
votes
1answer
39 views

Compute $\int_0^2\int_{\frac y2}^1e^{x^2}dxdy$

Compute $\displaystyle \int_0^2\int_{\frac y2}^1e^{x^2}dxdy$ $\displaystyle\int_0^2\frac {e^{x^2}}{2x}{\Big|}_{\frac y2}^1dy=\int_0^2\frac e2-\frac {e^\frac {y^2}{4}}{y}dy$. I am having trouble ...
3
votes
3answers
97 views

Evaluation of a particular type of integral involving logs and trigonometric function

Is there any closed form for $$ \int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$ if yes then how to prove it?
1
vote
1answer
45 views

Compute $\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle\int_0^1\int_1^2\frac{y}{x+y^2}dydx=\int_0^1\int_1^2y(x+y^2)^{-1}dydx$ How do I integrate the ...
0
votes
0answers
38 views

integral 2D involving complex exponential and cosine

I've some doubts about my solution of this integral: $$I(\phi_{1},\phi_{2})=\int_0^ {2\pi} \,d\phi_{1}\int_0^ {2\pi} \,d\phi_{2} \frac{e^{-in\phi_{1}} e^{-im\phi_{2}}}{2\pi}\frac{e^{il\phi_{1}} ...
2
votes
1answer
63 views

Solving exercise with Leibniz rule

I'm asked to prove that if $f(x) = \left(\displaystyle\int_0^x e^{-t^2}dt \right)^2$ and $g(x) = \displaystyle\int_0^1 \displaystyle\frac{e^{-x^2(t^2+1)}}{t^2+1}dt$ then $f'(x)+g'(x)=0$ and conclude ...
1
vote
1answer
93 views

Fubini Theorem. How it can be?

How this equality is true: $\left ( \int\limits_{-a}^a e^{-x^2} dx \right )\cdot \left ( \int\limits_{-a}^a e^{-y^2} dy \right )=\int\limits_{-a}^a \int\limits_{-a}^a e^{-(x^2+y^2)}\,dx\,dy$ For ...
2
votes
1answer
76 views

Evaluate $\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$

How can I evaluate the following double integral: $$\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$$ If we didn't have the restrictions $x<u, y<v$ polar coordinates would have ...
1
vote
1answer
78 views

Change the integration order of $\int^2_{-6} \int^{2-x}_{\frac{x^2}{4}- 1}f(x, y) \, dy\,dx$

This is a question from sample exam that I'm trying to solve but having difficults. Change the integration order of the integral: $$\int^2_{-6} \int^{2-x}_{\frac{x^2}{4}- 1}f(x, y) \, dy\,dx$$. ...
1
vote
1answer
65 views

Correct order of integration in spherical coordinates

I'm having troubles with an integral of the following form: $I=\int\limits_a^\infty dr\, r^2\int\limits_0^\pi d\theta\,\sin(\theta)f(r,\theta)$. My problem lies in the fact that these integrals are ...
0
votes
1answer
51 views

Scalar surface integral help.

$\iint xy \,\mathrm dS$ where $S$ is the surface of the tetrahedron with sides $z=0$, $y=0$, $x + z = 1$ and $x=y$. The answer is given as: $$\dfrac{3\sqrt{2}+5}{24}$$ ∫∫ xy dS = ∫∫ xy √(1 + ...
1
vote
1answer
64 views

Integration problem on multiple variables

I would like to ask help on this integration problem: $$\int_0^1\int_0^1\sqrt{x^2+y^2}\;\left[1-\alpha(1-2x)(1-2y)\right]dxdy$$ I was wondering if polar substitution is possible. I have done the ...
1
vote
1answer
62 views

Trying to understand Volume of a cone without the unit sphere

I have been working on the double integral proof for the volume of a cone. I found that I can use a unit-sphere Where the base of the cone is the equator and the height is the distance ${\rho}$ to ...
1
vote
0answers
40 views

An identity involving partial derivatives

Suppose $F(x,y)$ is a function of two variables satisfying $F(0,0)=0$. By differentiating some expressions, I obtained the identity $$ \frac{ \partial F}{\partial x}(x_0, y_0) = \int_0^1 ...
5
votes
4answers
126 views

Integrate: $\int_0^1 \mathrm{d}x_1 \int_0^1 \mathrm{d}x_2 \ldots \int_0^1 \mathrm{d}x_n \delta\left( \sum_{i=1}^n k_i x_i \right)$

Is there a closed form formula for this integral: $$\int_0^1 \mathrm{d}x_1 \int_0^1 \mathrm{d}x_2 \ldots \int_0^1 \mathrm{d}x_n \delta\left( \sum_{i=1}^n k_i x_i \right)$$ where $\delta(x)$ is the ...
2
votes
1answer
65 views

Triple Integral of region bounded by cylinder $x^2 + 3z^2 = 9$ and the planes $y = 0$ and $x + y = 3$

Here is the questoin with a diagram. My attempt at solution: $$x^2 + 3z^2 = 9 \Rightarrow 3z^z = 9-x^2 \Rightarrow z^2 = 3 - \frac{x^2}{3} $$ $$\Rightarrow -\sqrt{3 - \frac{x^2}{3}} \leq z \leq ...
0
votes
1answer
51 views

Find the volume of the region with triple integrals.

The volume of the region bounded by $y^2+z^2=1\ \text{and} \ z^2+x^2=1$ should be found. Cylindrical conditions are perhaps the most appropriate in this case but the limits of the integral are what ...
1
vote
1answer
35 views

Use spherical coordinates to evaluate

Use spherical coordinates to evaluate $\int_{-2} ^{2} \int_0 ^{\sqrt{4-y^2}} \int_{-\sqrt{4-x^2-y^2}} ^{\sqrt{4-x^2-y^2}} y^2\sqrt{x^2+y^2+z^2} dz \ dx \ dy$ I did like this. Is that right ? ...
2
votes
1answer
57 views

Find multiple integrals $I_{\max}(k,n)$ and $I_{\min}(k,n)$ in various ways

$I_{\max}(k,n)=\underbrace{\int\limits_0^1\int\limits_0^1\dots\int\limits_0^1}_k\left(\max\limits_{1\le i\le k}x_i\right)^n\,dx_1dx_2\dots dx_k$ ...
2
votes
3answers
84 views

Writing triple integrals in spherical coordinates over nonspherical/nonconical regions

Defining upper and lower limits of integration for $\rho$, $\theta$, and $\phi$ is relatively easy when writing a triple integral in spherical coordinates if the region of integration is defined by ...