0
votes
0answers
23 views

Use triple integrals to integrate over a tetrahedron

Integrate $f(x, y, z) = x^2 + y^2 - z$ over the tetrahedron with vertices $(0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3)$. I need to use triple integrals to solve this, so I made a diagram and set the ...
0
votes
1answer
52 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
0
votes
0answers
5 views

Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
3
votes
2answers
70 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
2
votes
2answers
21 views

Confusion about Spherical Coordinates Transformation

We have a function $$f(x,y,z) = \frac{e^{-x^2 -y^2 -z^2}}{\sqrt{x^2+y^2+z^2}}$$ and we want to integrate it over the whole $\mathbb{R}^3$. Then what i got is the following: $$\int_{\mathbb{R}^3}^ \! ...
0
votes
1answer
20 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
3
votes
1answer
43 views

Area of a Curved Surface

Find the area of the part o the surface $z=xy$ that lies within the cylinder $x^2+y^2=1$. I'm not sure how to set up the surface integral to compute this.
0
votes
1answer
54 views

Help with double integral

I need to prove if this integral exist (and some others) but i would like to know if there is a condition to say if the integral exist (for example in this case) that would help me solve this kind of ...
3
votes
1answer
45 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
0
votes
1answer
23 views

Area of a Paraboloid inside a Cylinder

Find the area of the part of the paraboloid $x=y^2+z^2$ that is inside the cylinder $y^2+z^2=9$. I'm not sure how to set up the integral to compute this. Thanks.
3
votes
2answers
41 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
12
votes
0answers
118 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 ...
2
votes
2answers
20 views

Question on Green's Theorem

Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$. Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where ...
2
votes
0answers
45 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
-2
votes
1answer
37 views

Find the average temperature for the following regions [on hold]

Consider the temperature function $T(x, y, z) = \large\frac{z}{1+x^2+y^2}$ where there is a heat source along the $z$ axis increasing in temperature as you get farther away from the origin. Find the ...
1
vote
4answers
41 views

Double integral with variable change, why the $2\pi$?

I've seen a lot of examples from my textbook where the result of an integration is $2\pi$ instead of $0$, as I would expect it to be. And several of my results will match the correct result if I ...
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 ...
0
votes
1answer
36 views

Line integral calculation

Problem Let $f:[-1,1] \to \mathbb R$ be a $C^1$ function such that $f(-1)=0=f(1)$ and $f>0$ on $(-1,1)$. Knowing that the graph of $f$ is contained in the set $\{(x,y) : x^2+y^2\leq 1, y\geq 0 ...
0
votes
1answer
22 views

Not getting limits of integration right (triple integrals)?

Calculate the volume bounded by $y+z=1$, $z=x^2-1$, $z=1-x^2$ and $y=0$ So our volume $V=\int \int \int _D1 dA$, all that's left is figuring out limits of integration. One method I saw was looking at ...
-4
votes
0answers
26 views

Triple intergrals with Volume [closed]

This question is a continuation of number 2 from the questions requiring deeper understanding of this homework. Suppose that you do not know the values of $a, b$, and $c$. Generalize what you did in ...
1
vote
1answer
21 views

Double integral calculation where $x=(y-1)^{2}-1$ and $y=x$. Not sure whether I should do it in terms of $y$ or $x$?

This is what it looks like: My first strategy was to separate it into two by drawing a vertical line at x=0 and calculate the first half in terms of x first, and the second half in terms of y ...
1
vote
1answer
72 views
1
vote
0answers
36 views

Question related to Stokes theorem proof

I was reading Stokes theorem proof from Apostol's Calculus II textbook and I got stuck at some parts of the proof. I'll copy the parts where I had doubts: Suppose $S$ is a surface parametrized by ...
2
votes
3answers
148 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 ...
1
vote
2answers
34 views

Double Integral Over a General Region

Evaluate the $\iint_R \sin\left(\frac{(x+y)}{2}\right)\cos\left(\frac{(x-y)}{2}\right) dA$ where R is a triangle with vertices $(0,0), (2,0), (1,1)$ using $u = \frac{(x+y)}{2}$ and $v = ...
2
votes
1answer
30 views

How do I set up a triple integral with vague parameters in the z coordinate direction?

The problem asks to find the volume of the solid bounded by $z = 16xy,~ z \geq 0,~ 0 \leq x \leq 5,~ 0 \leq y \leq 4$. But I am having trouble setting up the integral for the $z$ parameters. Any help ...
1
vote
0answers
25 views

A calculus problem regarding mass

I am reviewing calculus and working on the following problem. In $\mathbb{R}^{3}$, the density function is given by $\mu(x, y, z) = |z|$ and if the region $R$ is given by $R : 2 \leq x^{2} + y^{2} + ...
0
votes
2answers
20 views

Line integrals; How to set $t$ boundary?

I'm having a hard time understanding how to set t boundaries in line integrals... The question is: find the line integral of $f(x,y,z)$ over the straight line segment from $(1,2,3)$ to $(0,-1,1)$. I ...
0
votes
2answers
87 views

What is the difference between a line integral with respect to x or y and a Riemann integral with respect to x or y?

I'm finding the concept of line integrals with differentials including dx or dy hard to swallow intuitively. Specifically, I'm having trouble differentiating them from a Riemann integral. What are the ...
1
vote
3answers
43 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
1
vote
1answer
24 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
1
vote
1answer
57 views

Double integral over complicated region

Suppose we wanted to compute $\iint\frac {1}{1 + x^2 + y^2} dxdy$ over the region $\frac {(x-1)^2}4 + \frac {(y+2)^2}9 \leqslant 1$. It gets quite hairy if we use elliptical polar coordinates i.e. ...
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
1answer
20 views

Compute double integral on polar coordinates, find $r(\phi)$

I have the function $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ and the region $\{y\geq 2x^2-2, y\leq 3x\}$. The region is: To compute the integral in cartesian coordinates: ...
0
votes
1answer
41 views

Changing order of integration in cylindrical coordinates

I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The question is: Let D be the region bounded below by ...
1
vote
0answers
15 views

About integration limits over region

I need to integrate $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ over $\{y\geq 2x^2-2, y\leq 3x\}$ Im in doubt if it can be computed in just "one step" as: ...
0
votes
0answers
31 views

Reference Request: Fubini's theorem for non-negative functions

I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors ...
0
votes
2answers
22 views

Volume under surface

What is the volume under the surface $z = f(x,y) = x^4 + xy + y^3$ over the rectangle $R = [1,2] \times [0,2]$. I solved the double integral by integrating with respect to $x$ and then with respect ...
2
votes
0answers
61 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
2
votes
1answer
41 views

Divergence Theorem To Calculate Surface Integral

$M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}\leq z$} We are asked to find the surface area of this surface. This is my way: $\partial M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}= z$} so the ...
3
votes
2answers
89 views

Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?

If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is: $$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt. \qquad\text{(1)}$$ If $T=\{x=f(u,v); y=g(u,v)\}$ ...
1
vote
1answer
45 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
2
votes
2answers
60 views

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I'm having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried ...
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
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 ...
0
votes
3answers
48 views

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

I'm having a problem converting $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a circle on the x ...
0
votes
1answer
23 views

Integral with sum of differentials

Somebody can help telling what this integral means? $$\int (x+y) dx+dy$$ On the path $g(t)=(t,t^2)$ between 0 and 1
2
votes
1answer
68 views

How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
1
vote
2answers
36 views

solving double integrals

I'm trying to solve a double integral: $\displaystyle \int_{0}^{\frac{1}{2}}\int_{0}^{\frac{1}{2}-y}24xy\; dx \;dy$ I first solved in respect to $y$, making the $x$ a constant and plugged in the $y$ ...