2
votes
1answer
23 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
1
vote
1answer
42 views

Volume of the solid bounded by the planes (Checking the limits of the integral)

Find the volume $V$ of the solid bounded by the planes $x+y-z=3$ and $z=0$, and the cylinder $x^2+\frac{y^2}{4}=1$. My calculations give Polar $$V = \int_{\theta=0}^{\theta=\pi/2} \int_{r=0}^{r=1} ...
0
votes
2answers
40 views

Evaluate an integral using polar $\displaystyle\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx$

How do you evaluate the following integral using polar cordinates. $$\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\:\mathrm{d}y\:\mathrm{d}x$$ I converted it to polar coordinate making it ...
1
vote
2answers
87 views

Evaluate $\iiint xyz$

Evaluate $$\iiint_E xyz\, dV$$ where $E$ is the solid: $0\leq z\leq 9,\,0\leq y\leq z,\, 0\leq x \leq y.$ I am having a hard time drawing a picture of this solid $E$ to find out what the ...
0
votes
1answer
16 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
5
votes
1answer
62 views

Problem with a sequence with multiple integrals [duplicate]

How to compute the following limit, $\displaystyle \lim\limits_{n \to \infty} \int_0^1 \int_0^1 \ldots \int_0^1 \sin \bigg(\frac{x_1+x_2+\ldots+x_n}{n}\bigg)\,dx_1 \,dx_2 \ldots \,dx_n$ ? I will ...
0
votes
1answer
30 views

Calculate a triple integral - variable changed into spherical coordinates

The problem is to calculate $$\iiint_D x^2\,dx\,dy\,dz$$ where $D$ is determined by $x^2+2y^2+z^2\le2$. solution my attempt: why can I not do it like that? I change variables, calculate the ...
2
votes
2answers
47 views

Help changing the order of integration

So I need to change the order of integration. I am giving the following limits, $1 \leq x \leq 9$ and $\sqrt{x} \leq y \leq 4$. I am having no luck solving this one. Any help would be greatly ...
3
votes
2answers
47 views

Double integral help

I'm having difficulty with a question. It says By putting $x=r\cos(\theta), y=r\sin(\theta)$, prove that $$\int_0^{\infty}\int_0^{\infty}e^{-(x^2 + 2xy\cos(\alpha)+y^2)}dx\ ...
0
votes
1answer
19 views

Solving a double integral using substitution

The problem: Evaluate $$\iint_{D}(x+y)^2(x-y)^5\:\mathrm{d}x\:\mathrm{d}y,$$ where $D$ is a rectangle with vertices in $(0, 1), (1, 0), (1, 2), (2, 1)$. So I drew the square and thought up this ...
1
vote
2answers
41 views

Aside from this two practical technique to compute any integral, what else? [on hold]

Aside from this two practical technique to compute any integral, what else could called a fundamental method but not approximate method like Riemann Sum? These two method I've been referring to are ...
1
vote
1answer
21 views

Calculating the center of mass in spherical coordinates

So normally, to calculate the center of mass you would use a triple integral. In my particular problem, I need to calculate the center of mass of an eight of a sphere where it's density is ...
1
vote
1answer
30 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
0
votes
0answers
18 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
0
votes
2answers
46 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
2
votes
0answers
41 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
0
votes
2answers
28 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
3answers
24 views

Simple Line Integral — Just making sure I'm right

Compute the line integral $\int_C xe^{z^2} ds$ where C is the piecewise linear path from $(0,0,1)$ to $(0,2,0)$ to $(1,1,1)$. I started out by parametrizing the curve and got: $C_1: x = 0,\space y ...
0
votes
0answers
29 views

Line Integral over triangle/force field

having some trouble with some line integrals: Compute the line integral of $F = <e^z, e^{x-y}, e^y>$ over the path from $A(2,0,0)$ to $B(0,4,0)$ to $C(0,0,6)$ to $A$. Thanks!
-3
votes
0answers
14 views

Line integral FTLI

Suppose that $\nabla f(x,y,z)=2xyze^{x^2} {\textbf{i}} + ze^{x^2} {\textbf{j}} \hat{\jmath} + ye^{x^2} {\textbf{k}}$ if $f(0,0,0)=-2$ Find $f(3,3,5)$. I tried many different ways but I still could ...
1
vote
1answer
24 views

Line integral segment of parabola

Suppose $$ \vec{F} = \nabla f(x,y) = 6y \sin (xy) \vec{i} + 6x \sin (xy) \vec{j}, $$ and C is the segment of the parabola $y = 5 x^2$ from the point $(2,20)$ to $(6,180)$. Then, what is $$\int_C ...
3
votes
2answers
166 views

Green's first identity

Good morning/evening to everybody. I'm interested in proving this proposition from the Green's first identity, which reads that, for any sufficiently differentiable vector field $\mathbf{\Gamma}$ and ...
-1
votes
1answer
37 views

How to evaluate this integral step by step?

Evaluate $$\int\int\sin(x-y)dxdy$$ Is it difficult? see: https://www.wolframalpha.com/input/?i=int+int+sin%28x-y%29+dx+dy
0
votes
1answer
11 views

Line integral for three line segments

Find where C consists of the three line segments from (2,0,0) to (2,1,0) to (0,1,0) to (0,1,5). I tried to find it like the other line integral but it seems that there is something wrong!
1
vote
0answers
11 views

Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
0
votes
1answer
19 views

Evaluate $I=\int \int _{\delta}(x^my-y^nx)dA$,if $m,n \in \mathbb{N}$ and $\delta$ is the part of the unit disc in the upper half-plane

using polar cordinates i get the following result,by considering a circle $x^2+y^2\le 1$ and $y\ge 0$ $I=\int_{0}^{\pi}\int_{0}^1 (r^m\cos^m\theta r\sin \theta-r^n\sin^n\theta r\sin \theta \cos ...
0
votes
3answers
52 views

$x^2 + y^2 - y = 0$ is… a cylinder?

I've this question: Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$. Is this second equation even a closed shape? If one were to ...
0
votes
1answer
32 views

help me to find the Line Integral

Find for on the curve counterclockwise around the unit circle C starting at the point (1,0). I dont know the way to do that I tried many ways but I still could not get the right answer
0
votes
1answer
7 views

find the line integral

Evaluate the line integral where C is the straight line path from (2,3) to (7, 5). I dont know the way to do that I tried many ways but I still could not get the right answer
0
votes
1answer
10 views

Evaluate the Line Integral

Evaluate the line integralwhere and C is given by the vector function , That should be a simple question but I'm getting a wrong answer! The way I did it is evaluating the integral with t instate of ...
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 ...
1
vote
2answers
39 views

Is it possible to find a function if we know its differential?

Not something we were taught at uni yet, just something that peaked my curiosity. If I was given a derivative of a scalar function, for example $f'(x)=x$ then I know that $f(x)=\frac{x^2}{2}$ (let's ...
1
vote
1answer
58 views

Integration w/ Change of Variables

folks. I've got this question: Let $D$ be the region $\{(x,y) ~|~ 0 \leq y \leq x, 0 \leq x \leq 1\}$. Evaluate: $$\iint_D (x + y) dxdy$$ by making the change of variables $x = u + v$, $y = u ...
1
vote
1answer
33 views

Volume of the Region bounded by $y = 2x^2 +2z^2$ and the plane $y=8$

I have the find the volume of the region bounded by the paraboloid $y = 2x^2 +2z^2$ and the plane $y=8$. Is the volume (using triple integrals) just ...
0
votes
0answers
91 views

Volume enclosed by two spheres (triple integral, cylindrical coordinates)

The question: Find the volume of the solid enclosed by the sphere $x^2 + y^2 + z^2 - 6z = 0$ , and the hemisphere $x^2 + y^2 + z^2 = 49 , z ≥ 0$ I set up the triple integral ...
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, $$\iiint_{G} \nabla \cdot \vec{F} \, dV = \iint_{\sigma} \vec{F} \cdot \vec{n} \, dS$$ ...
0
votes
2answers
77 views

Double integral notation

Over a region D (a bounded, closed and connected region), can we write the double integral $\iint\limits_D \, f(x,y)\,dx\,dy$ as $\iint\limits_D \, f(x,y)\,dy\,dx$ (note the order of $dx$ and $dy$)?
0
votes
1answer
35 views

Double integral - domain of integration

Calculate the following double integral: $ \int \int_S x\; dx \; dy $, where $S$ is the region limited by the circle of radius 1 and center (0,1) and the line that pass through the points (2,0) and ...
0
votes
1answer
18 views

Triple integral - wedge shaped solid

Find the volume of of the wedge shaped solid that lies above the xy plane, below the $z=x$ plane and within the cylinder $x^2+y^2 = 4$. I'm having serious trouble picturing this. I think the z ...
1
vote
1answer
29 views

Polar equation — find area under graph using double integral

What is the area of the region in the plane bounded by the curve given in polar coordinates $r = 4 + 2\cos(2\theta)$? Could someone walk me through the conversion of polar coordinates to rectangular ...
0
votes
1answer
45 views

Z coordinates of a solid object moving

The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density at the point (x,y,z) and occupies a region W, ...
0
votes
2answers
24 views

what are the spherical coordinates

What are the spherical coordinates of the point whose rectangular coordinates are (3 , 1 , 4 ) ? I got that =sqrt26 but I could not find the values for the others
0
votes
2answers
25 views

help me to find the triple integral

Use cylindrical coordinates to calculate for the given function and region: I found that the limits are for x 0 to 2pi r 0 to 5 and z from r^2 to 25 and the integration function z*r. I got ...
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
36 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 ...
0
votes
0answers
40 views

Mass of the cylinder

The density,, of the cylinder varies with the distance, r, from the z-axis: Find the mass of the cylinder, assuming x,y,z are in cm.
0
votes
1answer
17 views

spherical coordinates to find the triple integral

Use spherical coordinates to evaluate the triple integral where E is the region bounded by the spheres and
1
vote
1answer
31 views

Volume within the sphere

Find the volume of the solid that lies within the sphere , above the xy plane, and outside the cone My problem is finding the integral function and the limits
1
vote
1answer
42 views

please help me to do Triple integral

Integrate over the region in the first octant above the parabolic cylinder and below the paraboloid I could not get the limits right even that I tried many one but I still could not get it
1
vote
1answer
34 views

Triple Integral

Use cylindrical coordinates to evaluate the triple integral $$\iiint_{\mathrm{E}}\sqrt{x^{2}+y^{2}}\, dV,$$ where $\mathrm{E}$ is the solid bounded by the circular paraboloid $z=1-9(x^{2}+y^{2})$ and ...