-2
votes
0answers
13 views

how to increace the volume to a specific volume in revolution of solid, using integration [on hold]

two functions, f(x)= 1/9(x-2)^2+7 domain range:{0,10}, g(x)=1/7(x-5)+0.7 domain range: {10,13} increase the volume to 1000ml to 1050mL using integration.
3
votes
3answers
63 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
37 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 ...
2
votes
3answers
36 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
84 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
58 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
91 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
51 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
24 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
79 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
32 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
26 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
20 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
45 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} ...
0
votes
1answer
66 views

Splitting Integral into Two Parts

This question might seem very simple, but I can't seem to figure it out. Suppose I have an integral over a square region. I was wondering in which case it would be incorrect to split the integral into ...
0
votes
1answer
25 views

Vectorial Calculus proof [closed]

Please help me to prove the following identity wherein $\phi$ is a scalar field and d$\vec l$ is the linear element: $$\int \nabla \phi \cdot d\vec l = \int d\phi$$ hopefully step by step. ...
0
votes
1answer
67 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
0
votes
1answer
35 views

Finding a closed line integral using Stokes' Theorem

Find the line integral $\int_C \vec{F} \cdot \vec{dS}$, where $C$ is the circle of radius 3 in the $xz$-plane oriented counter-clockwise when looking from the points $(0, 1, 0)$ into the plane and ...
-1
votes
2answers
45 views

Evaluate the flux integral [closed]

Evaluate the flux integral $$ \int\!\!\int_{S} {\rm curl\left(\vec{F}\right)} \cdot \vec{dS} $$ where $$ \vec{\rm F}(x, y, z) =\langle xe^{y^2}z^3 + 2xyze^{x^2 + z}, x + z^2e^{x^2 + z}, ye^{x^2+z} + ...
-1
votes
0answers
19 views

Transformations of variables

I don't know how apply the second transformation of variables in this integral. The integrated function can be assumed to be $N_{MJ}(\bar\xi)*D^{-1}(\bar\xi)*\bar\xi_i*\bar\xi_n=\bar\xi_1* \bar\xi ...
-1
votes
1answer
35 views

Find finite area between curves [closed]

Find the finite area enclosed between $r= a \sin 4(\theta)$ and $r= a \sin 2(\theta)$ in polar coordinate system.
0
votes
2answers
59 views

Using Stokes theorem to find the integral of a vector field over the curve of intersection of two surfaces

Find $\int_C{ \vec{F} \cdot \vec{dr}},$ where $F(x, y, z) = \langle 2 x^2 y , 2 x^3 /3, 2xy\rangle$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z = y^2 - x^2$ and the ...
0
votes
1answer
42 views

Find $\int_0^1 \int_{3x}^3 (x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$ [closed]

You can use a calculator after simplification if its not possible by hand All Ideas will be appreciated Also If you could find $$\int_0^1 \int_{3x}^3 x(x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$$ ...
0
votes
1answer
23 views

multivariable calculus double integration volume question

Use a double integral to find the volume of the solid bounded by graphs of the equations given by: $$\begin{align}z=xy^2, \text{ where: } &z>0\\&x>0\\&5x<y<2\end{align}$$ My ...
2
votes
3answers
76 views

Find $\iiint_E (1-x^2-2y^2-3z^2)~\mathrm{d}V$, where $E$ is the region inside the ellipsoid $x^2+2y^2+3z^2=1$ [closed]

My textbook asked to use a calculator to find this. Not sure how to setup the triple Integral.
6
votes
3answers
208 views

find $ \int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$

I am looking for an approximation to the nearest integer of $$ \int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz.$$ Wolfram alpha gives up and says "computation time exceeded". I tried, ...
1
vote
0answers
28 views

Evaluating a path integral without being given a parametrization

Find the mass of the wire formed by the intersection of the sphere $x^2 + y^2 + z^2 = 1$ and the plane $x + z = 0$ if the density of the wire is $6y^2$ grams per unit length. I am completely stuck on ...
0
votes
2answers
49 views

Calculations of double integral of $xy^2$ over the region between $y=0$ and $y=4-x^2$

Find $\iint_D f~\mathrm{d}A $ where $f(x,y) = xy^2$. Region $D$ is between $y=0$ and $y=4-x^2$ If you draw the graph, you can see that it equal to $0$, however if you calculate the answer ...
0
votes
0answers
18 views

Help with calculating line integrals and potential functions [duplicate]

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
-1
votes
0answers
37 views

potential functions and line integrals

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
1
vote
1answer
75 views

line integral…

Calculate $$\int_Γ f \, d\ell$$ for $f(x,y) = y, \; y=x^{1/2}$, $ x $ is in $[2,6]$. I know (now) that it means that: $$\int_\Gamma f \, d\ell=\int_a^b f(\Gamma(t)) \cdot \|\dot\Gamma(t)\| \, dt$$ ...
0
votes
0answers
19 views

Simplification of an integral comprising of vector-variables

How can I evaluate the simplify the integral $\int \rho (\bf{r^{\prime}})\, \delta (\frac{\sigma}{2}-r) d \bf{r^{\prime}}$ where $\delta$ is the dirac-delta function given that $\rho$ is constant ...
0
votes
2answers
31 views

How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
1
vote
1answer
26 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
2
votes
2answers
30 views

Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
1
vote
1answer
35 views

Work done by a force field line integrals

Find the work done by the force field $F(x, y) = \langle 2x \sin(y), 2y \rangle$ on a particle that moves along the parabola $y = x^2$ from $(-1, 1)$ to $(2, 4)$. So to use line integrals to solve ...
1
vote
1answer
62 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
2
votes
1answer
43 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
1
vote
1answer
35 views

Finding a function from a vector field

The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...
3
votes
2answers
243 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
62 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
1answer
18 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
0
votes
1answer
43 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 ...
1
vote
1answer
69 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
1answer
13 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 ...