0
votes
0answers
8 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 $z=- \sqrt{9-x^2 - y^2}$ ...
0
votes
0answers
16 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 rewrote ...
1
vote
1answer
31 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, \ ...
3
votes
2answers
87 views
+50

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
41 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
97 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 ...
2
votes
0answers
29 views

To find the volume dilation, integrate the determinant of the Jacobian

On the road toward proving the change of variables theorem in several variables, is there a painless way to show that $$\text{Vol}(\phi(U))=\int_{U}|\text{det}(d\phi)|,$$ where $\phi$ is $C^1$, ...
1
vote
1answer
34 views

Line integral exercise

Let $f:[-1,1] \to \mathbb R$ be a $C^1$ function such that $f(-1)=f(1)=0$ and $f>0$ in $(-1,1)$. Knowing that the graph of $f$ is containted in the semicircle $x^2+y^2 \leq 1$, $y \geq 0$, ...
1
vote
1answer
46 views

Stokes' Theorem problem (right triangle)

I am asked to demonstrate the truth of Stokes' Theorem ($\int_T curl(\vec{v}) \cdot \vec{da} = \int_{\partial T} \vec{v} \cdot \vec{dl}$) in the following problem/case: Let $\vec{v} = x y \hat{x} + ...
1
vote
3answers
43 views

triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
1
vote
1answer
37 views

Multiple integral what am i doing wrong?

$$ \iint _d (x-y)^2*{{{e^y}^+}^z} \qquad d:x-y\ge -1,x-y\le1,x+y\ge1,x+y\le3 $$ i was trying to separate it to two multiple integral integrals but i cant integrate this integral, i was trying to ...
1
vote
0answers
15 views

Triple integral over a region bounded by surfaces.

The problem is to calculate $$I=\underset{S}{\iiint}x^2\;dV,$$ where $S$ is the region bounded by $x+y^2=1$, $x+z^2=1$ and the plane $zy$. In my conclusions (that follows from this reasoning and ...
1
vote
0answers
21 views

Parametrization of a Closed Section of a Sphere

I'm trying to verify Divergence theorem for a specific vector field through the part of a sphere in the first octant. I've done the volume integral, I'm just having trouble parametrizing the surface ...
0
votes
1answer
16 views

Line integral of vector field

Let $F$ be the vector field over $\mathbb R^3$ given by $$F(x,y,z)=(4x(1+\frac{z^2}{8}),\frac{\pi}{3}\cos(\frac{\pi}{3}y),\frac{x^2z}{2}-e^{\frac{z}{3}})$$ and let $C$ be the curve parametrized by ...
0
votes
0answers
46 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
0
votes
0answers
34 views

Testing Divergence Theorem using Spherical Coordinates

I'm trying to verify the divergence theorem using spherical coordinates for the vector field $\vec{F}=r^2cos^2\theta(cos\theta\hat r-sin\theta\hat\theta)$ through the top half of the unit sphere. ...
0
votes
0answers
17 views

Integrating a three variable function

Let $\Omega$ be the region in the $xy$ plane bounded by a quarter circle of radius $a$, a straight line of slope $-1$ from $(0,b)$ to $(b,0)$, and the coordinate axes. Now consider the ...
3
votes
1answer
37 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
1
vote
1answer
59 views

Evaluate line integral without parameterizarion

It's been brought to my attention that line/surface integrals and integrals of differential forms in general can be evaluated without introducing a parameterization, however I haven't been able to ...
0
votes
0answers
24 views

Integrating differential forms over a box

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
3
votes
1answer
61 views

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
1
vote
2answers
60 views

Find the volume below $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ in the first quadrant

I understand that we have to use transformation $$x = u^2, y = v^2, z = w^2$$ but I cannot figure out the limits. I just need a rough sketch of how to approach this. Could anyone give me some ideas?
2
votes
0answers
23 views

Normal Vector Affecting The Divergence Theorem

$\newcommand{\Div}{\operatorname{Div}}$I'm going to use an example to explain what I'm trying to ask. Let $T =\{(x,y,z): x^2+y^2=z^2, 0\leq z\leq3\}$, I'm asked to calculate $\iint_T ...
1
vote
1answer
14 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
2
votes
1answer
49 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
0
votes
1answer
44 views

Integral of a bivariate normal cdf

Let $$ \Phi_2(x,y;\rho):=\int_{-\infty}^y\int_{-\infty}^x \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(s^2+t^2-2st\rho)} \, ds \, dt $$ be the joint cdf of bi-variate normal random ...
2
votes
2answers
50 views

Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...
1
vote
1answer
25 views

Calculating moment of inertia about the $z$-axis of solid with constant density

I have the following math problem: Find the moment of inertia about the $z$-axis of the solid in the first octant that is bounded by the coordinate planes and the graph of $x+y+z=1$ if the density ...
3
votes
3answers
77 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
0
votes
2answers
39 views

Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
8
votes
2answers
235 views

How find this integral $I=\int_{0}^{1}\int_{0}^{1}\frac{\ln{(1+xy)}}{1-xy}dxdy$

Find this integral $$I=\int_{0}^{1}\int_{0}^{1}\dfrac{\ln{(1+xy)}}{1-xy}dxdy$$ My try: since $$\dfrac{1}{1-xy}=\sum_{n=0}^{\infty}(xy)^n$$ so ...
2
votes
3answers
39 views

Finding the partial derivatives of $h(x)=\int_{0}^{\|x\|} f(t)\, dt$

Find the partial derivatives of $$h(x_1,\dots,x_n)=\int_{0}^{\|x\|} f(t) dt$$ where $\|x\|$ is the Euclidean norm of $x=(x_1,\dots,x_n)$ and $f$ is some continuous function. I'm sorry but I'm really ...
2
votes
2answers
93 views

Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
0
votes
3answers
63 views

Area of the region: $\;x ≥ 0; \;−x\sqrt3 ≤ y ≤ x\sqrt3;\,\;(x−1)^2 + y^2 ≤ 1$.

Can anyone please explain how to set up the needed integral? I need to calculate the area of the region given by: $x ≥ 0,$ $-x\sqrt3 ≤ y ≤ x\sqrt3,$ $(x−1)^2 + y^2 ≤ 1$.
2
votes
1answer
62 views

What is the meaning of $d\vec S$ in a surface integral?

Can someone explain if I have a surface $z= 9-x^2-y^2$ What would $\vec{n}$ be? What would $d\vec{S}$ be? Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!
1
vote
2answers
92 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
8
votes
1answer
56 views

Changing the order of integration without sketching?

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...
1
vote
1answer
43 views

Finding the region of integration

Let $A$ be the directed line from $(-1,-1)$ to $(1,1)$ and $B$ be the curve which starts at (1,1) and moves along $x^2+y^2=2$ up to $(-1,-1)$. Let $C$ be the union of $A$ and $B$ and $R$ be the region ...
1
vote
0answers
25 views

Showing that $f\in C'(\mathbb{R^2},\mathbb{R})$

Let $$f(x,y)=xy\int_{x^2-y^2}^{x^2+y^2}e^{\cos(xyt)}dt.$$ Prove that $f\in C'(\mathbb{R^2},\mathbb{R})$. I'm not exactly sure how to approach this problem. Here's what I've tried: First I ...
1
vote
2answers
28 views

Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
1
vote
2answers
82 views

Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$

I need the approximation to four decimals Not sure how to start or if a closed form solution exists All Ideas are appreciated
0
votes
1answer
18 views

Change of variables theorem in the case $L^1_{\textrm{loc}}(U)$?

I'm trying to write a version of the change of variable theorem for the case of locally integrable functions on open subsets of $\mathbb R^n$. Statement: Let $U, V\subseteq \mathbb R^n$ be open sets ...
1
vote
0answers
37 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
1
vote
1answer
32 views

Surface integral on sphere

Is there a direct way to calculate the surface integral of the gradient of some smooth function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ over the sphere $S^2$ without knowing $f$? $$ ...
0
votes
2answers
23 views

Marginal density function question

The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, ...
1
vote
2answers
30 views

Find the centroid of the boomarang shaped region for the parabolas $y^2=-4(x-1)$ and $y^2=-2(x-2)$

I know the formulas, I only need assistance setting up the initial integral. So my order of integration must be $\mathrm{d}x$ $\mathrm{d}y$. Then if we solve the parabola for $x$ the new integral we ...
0
votes
1answer
47 views

Use the transformation $x=u^2-v^2$, $y=2uv$ to evaluate the integral

$$\int_0^1 \int_0^{2\sqrt{1-x}} \! \sqrt{x^2+y^2} \, \mathrm{d}y\,\mathrm{d}x$$ Here's where I'm at: $J(x,y)=4u^2+4v^2$ Substituting $x$ and $y$ into the integral: $\sqrt{(u^2-v^2)^2+4u^2v^2} ...