All Questions
127
questions
1
vote
1
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42
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Evaluate $\iint_S\vec{F}$ where $S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above and below by two planes.
Use the divergence theorem to evaluate $$\iint_S\vec{F}$$ where $\vec F(x,y,z)=\langle xy^2,x^2y,y \rangle$ and $\vec S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above ...
- 3,065
3
votes
1
answer
64
views
Find volume of body between surfaces
Problem: Find volume of body defined as follows:
$z^2=xy$, $(\frac{x^2}{2}+\frac{y^2}{3})^4=\frac{xy}{\sqrt{6}}$, $x, y, z \ge 0$.
My solution:
So we're working in the all positive octant of the ...
- 41
3
votes
2
answers
81
views
Contradiction (or mistake) in $\displaystyle \iint x^y ~ dx ~ dy$
I came across this double integral at an Instagram post and I tried to solve it. The integral in question is the following:
$$I = \iint x^y ~ dx ~ dy$$
So, I begun by calculating the inner integral ...
- 99
0
votes
0
answers
44
views
What is the work done along a quarter circle in a vector field?
What is the work done along a quarter circle in a particular vector field?
To better understand different types of line integrals, I set up the problem below, and request verification:
Let $F$ be a ...
- 4,450
1
vote
0
answers
149
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How to use the hint by the author? (Problem 3-23 in "Calculus on Manifolds" by Michael Spivak.)
Problem 3-23.
Let $C\subset A\times B$ be a set of content $0$. 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 $0$. Show that $A'$ is a set of measure ...
- 1,138
1
vote
0
answers
179
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Problem 3-21 in "Calculus on Manifolds" by Michael Spivak.
The following problem is Problem 3-21 in "Calculus on Manifolds" by Michael Spivak.
I solved this problem but I am not sure if my solution is right. Is my solution right? If so, how to ...
- 1,138
0
votes
1
answer
45
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Compute the double integral $\iint_{(x,y):0<\tfrac{x}{2}\leq{y}\leq\tfrac{3x}{2}<1}dydx$
Attempt: I sketched the region, and decided it would be easiest to break up the area into two integrals $A_1,A_2$ and sum. Is it correct that $A_1,A_2$ are Type-I integration regions? Is there a more ...
- 1,173
0
votes
0
answers
47
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Calculate $\int_\phi \omega$, where: $\phi(t)$ and $\omega$ are specified.
Calculate $\int_\phi \omega$, where:
$\phi(t): [0,1] \to \mathbb{R}^3$
$\phi(t) = \Big( 2 + 16t^2(t-1)^2, 4t(1-t), (1-t) \big( 1 + 16t^2(t-1)(2t-1) \big) \Big)$
$\omega = \big( 2xz^2 e^{x^2 + y^2} + ...
- 1,086
2
votes
2
answers
64
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Compute $\int^2_0\int^2_y \frac{28}{3}(x^2+xy)dxdy$
\begin{align*}
\int^2_0\int^2_y \frac{28}{3}(x^2+xy)dxdy&=\frac{28}{3}\int^2_0\int^2_y x^2dxdy+\frac{28}{3}\int^2_0\int^2_y xydxdy\\
&=\frac{28}{3}\int^2_0\left[\frac{x^3}{3}\right]^2_ydx+\...
- 1,426
2
votes
0
answers
61
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Integrating monomials over $\mathbb{S}^{n-1}$ - Folland
This is an exercise from Folland: Suppose $f(x)=\prod_{j=1}^n x_j^{\alpha_j}$ where all $\alpha_j$ are even. Then show that
$$\int_{\mathbb{S}^{n-1}} f d \sigma= \frac{2 \prod_{j=1}^{n}\Gamma(\beta_j)}...
- 1,777
1
vote
1
answer
63
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Find the moment of inertia of the wire about the y-axis
This is question 10 from Exercise 8b from A Second Course in Mathematical Analysis (J. C. Burkill, H. Burkill).
A wire is shaped in the form of a twisted cubic given by $$\mathbf{r}(t) = (3t - t^3, ...
- 759
1
vote
1
answer
46
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(Solution verification) Find the integral of the function $f(x,y,z)=xy$ over the region $R$ bounded by $x^2 + y^2=1, z=0, z=1, x\geq 0 , y \geq 0$
Find the integral of the function $f(x,y,z)=xy$ over the region $R$ bounded by $x^2 + y^2=1, z=0, z=1, x\geq 0 , y \geq 0$
Attempt: It is quite apparent we ought to use cylindrical coordinates. In ...
- 516
1
vote
1
answer
55
views
(Solution verification) Compute the integral $\iint_{R}\frac{1}{(1+x+y)^2}dA $ if $R$ is the region bounded by lines $y=2x$,$x=2y$ and $x+y=6$
Compute the integral
$\iint_{R}\frac{1}{(1+x+y)^2}dA $ if $R$ is the region bounded by lines $y=2x$,$x=2y$ and $x+y=6$
Essentially I only need to know if I got the integration bounds correctly. I ...
- 516
2
votes
0
answers
59
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$\int_W x^2y^2 = \int_V x^2y^2$. Is my proof of this equation right? ("Analysis on Manifolds" by James R. Munkres)
I am reading "Analysis on Manifolds" by James R. Munkres.
The following example is EXAMPLE 4 on p.149:
EXAMPLE 4. Suppose we wish to integrate the same function $x^2y^2$ over the open set $$...
- 8,740
1
vote
1
answer
125
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Wrong answer key in Thomas' Calculus 14th edition?
The below exercise is from the textbook I'm reading and I think the answer given at the back of the book is wrong.
Integrate
$$
\iint_R xydxdy
$$
where $R$ is the region bounded by $x=2a$ and $x^2=...
- 1,156