0
votes
2answers
35 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 ...
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
2answers
77 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
65 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
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} + ...
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 $$\int_0^1 \int_{3x}^3 x(x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$$ ...
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, ...
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
61 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 ...
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 ...
4
votes
2answers
87 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 ...
29
votes
2answers
588 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 ...
1
vote
2answers
75 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 ...
1
vote
1answer
75 views
2
votes
3answers
151 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 ...
0
votes
1answer
27 views

exchanging partial derivative and an integral

Does $\frac{\partial }{\partial x}\int_{u(x)}^{v(x)}f(x,t)dt$ = $\int_{u(x)}^{v(x)}\frac{\partial }{\partial x}f(x,t)dt$ ?? If yes, how did we do that although there are 2 functions $u$ and $v$ ...
1
vote
0answers
23 views

Integrals depending upon a parameter

There was an exercise, in my professor's book, asking to prove the continuity of an integral depending upon a parameter. Namely, the hypothesis were: Let $D$ be a measurable subset of $\mathbb{R}^n$, ...
1
vote
1answer
50 views
0
votes
3answers
64 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
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
1answer
34 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
0
votes
2answers
46 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 ...
1
vote
0answers
41 views

Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...
0
votes
1answer
40 views

Use divergence theorem to find $\iint_S (2x+2y+z^2) dS$ Where $S$ is the sphere $ x^2+y^2+z^2 = 1$

I tried a lot but it gets ugly really soon, any help will be greatly appreciated. T hanks
3
votes
1answer
60 views

Interesting dilemma, answer not matching with stewart, My work is Included

Question : Compute flux through the upper hemisphere of $x^2+y^2+z^2 = 1$ . Where $$\textbf{F} = \left( z^2x\right)\textbf{ i }+\left[\dfrac{1}{3}y^3+ \tan z\right]\textbf{ j } + \left(x^2z+y^2 ...
0
votes
3answers
44 views

Triple integral calculation gone wrong?

Calculate the volume of the solid bounded $K$ by $$z \geqslant x^2 + y^2 - 1, \quad z \leqslant \sqrt{x^2+y^2} + 1, \quad z \geqslant 0$$ The triple integral I set up is $$ \iiint_K ...
2
votes
1answer
44 views

Calculate $\int_D x^3y\ dx\,dy$

Let $D$ the bounded region by the $y$-axis and the parable $x= -4y^2 + 3$. How can I calculate the integral $$\int_D x^3y\ dx\,dy$$ I am stuck with this problem some help to solve this please.
1
vote
2answers
43 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
2
votes
1answer
46 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
0
votes
1answer
54 views

Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Prove without using the divergence theorem. The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem. Thanks in advance.
3
votes
3answers
84 views

calculate $\int_{0}^{\pi} \int_{0}^{x}\log(\sin(x-y))dydx$

I was asked to find the integral $\iint_A \log(\sin(x-y))dxdy$ where $A$ is the triangle $y=0, x=\pi, y=x$ in the first quadrant. I was given a hint: evaluate $\int_{0}^{\pi}\log(\sin(t))dt$ using ...
2
votes
4answers
63 views

Mass of elipsoid

We are given the elipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \leq 1$ with density function $p(x,y,z)=x^2+y^2+z^2$. Find the mass of the elipsoid. what I did: I used the ...
0
votes
2answers
39 views

how do we interpret this integral from polar co-ordinates

$$\text{Find } \int_C rdr$$ Where $C$ is any closed loop. I feel that the answer is zero, i have no hard reasoning. Here $r$ is the parameter from the polar coordinates.
1
vote
3answers
105 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
1
vote
1answer
37 views

Integrate $f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$

$f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$ I have to integrate it over $E=[0,1]\times[0,1]$ I can integrate the inner integral ...
1
vote
0answers
39 views

Imaginary part of a triple intragral

I'm only an introductory calculus student(I just finished my high school AP calculus BC course) so I apologize if these questions seem a bit elementary. I'm using Mathematica to perform a triple ...
0
votes
2answers
40 views

How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
2
votes
1answer
35 views

Stokes theorem: $ \vec{v} = (xz, -y, x^2y)$

Evaluate $$ \iint_S \nabla \times \vec{v} . \vec{N} dS$$ where $\vec{v} = (xz, -y, x^2y)$ and S consists of the 5 faces, not on xy plane, of the cube$ [0,2] \times [0,2] \times [0,2] $ and $ ...
0
votes
0answers
13 views

Gauss theorem: $\vec{v} = (x^2 + ye^z, y^2 + ze^x , z^2 + x e^y)$

If D is the region bounded by the cylinder $x^2 + y^2 = 1, z=0$ and $z= x+2$, use the Gauss Theorem to evaluate $$ \iint_S \vec{v} . \vec{n} \; dS$$ where $S$ is $\partial D$ , $\vec{n} $ points to ...
0
votes
3answers
35 views

Surface Integral of $\frac{\vec{r}}{|\vec{r}|^3}$

Let $$\vec{v} = \frac{\vec{r}}{|\vec{r}|^3},$$ $\vec{r}=(x,y,z)$. Evaluate $$ \iint\limits_S \vec{v} \cdot \vec{n} \, dS$$ with $\vec{n} $ pointing to the exterior of the surface $S$ ...
0
votes
1answer
26 views

Surface integral over a rectangle

Evaluate $$ \iint_S \vec{v} . \vec{n} dS$$ for $\vec{v} = (x+y, -2y - 1, z)$ and $S$ the rectangle of vertices $(1,0,1),(1,0,0),(0,1,0),(0,1,1)$ and $\vec{n}$ points in the opposite direction of ...
0
votes
1answer
23 views

Surface integral over the plane $x+y+z=2$

Evaluate $$\iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy$$ where $S$ is the part of the plan $x+y+z=2$ in the first octant, with normal $n$ such that $n . (0,1,0) \geq 0$ My ...
0
votes
1answer
37 views

Surface integral over a sphere - parametrization

Evaluate the surface integral of the field $A(x,y,z)=(xy, yz, x^2)$ over the sphere $S$ givn by $x^2 + y^2 + z^2$ with the normal vector pointing to the exterior of the sphere I've tried doing this ...
3
votes
1answer
63 views

approximate this fancy looking double integral

$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
2
votes
1answer
88 views

This double integral

$$ \int_0^1\int_0^1x^3y^2\sqrt{1+x^2+y^2}\hspace{1mm}dxdy$$ We have to compute this up to 4 decimal places