All Questions
Tagged with integration multivariable-calculus
5,463
questions
2
votes
1
answer
32
views
Expressing integral based on the change of variables in double integral
Let B be the region in the first quadrant bounded by the curves $xy=1$,$xy=3$,$x^2−y^2=1$, and $x^2−y^2=4.$
Evaluate $\iint_{B} (x^2 + y^2 ) dx dy$ using the change of
variables $u=x^2−y^2$,$v=xy.$
...
0
votes
0
answers
29
views
How can we calculate the Integral of a vector norm along a parametrized line?
I feel a bit silly because I've been trying to calculate the following integral and I believe that the solution is likely extremely easy, but I am blanking. Let $p \in (0,1)$, Given two vectors $\xi$ ...
0
votes
0
answers
37
views
Is this calculus derivation process correct?
This is part of the economics romer model, finding the Lagrangian value
$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\
s.t.\int_{i=0}^{A}L(i)^{\phi}di=1
$...
- 101
3
votes
1
answer
323
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Flux calculation - what did I do wrong?
The exercise asks to calculate the flux of $\mathbf{F}=(4x,4y,z^2)$ through the surface $x^2+y^2=25, 0\le z \le 2$
I calculated using Gauss' theorem and I obtained $500\pi$, which is, as the teacher ...
- 185
0
votes
0
answers
36
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Integration measure for a strange substitution
I have a 2D integral over a momentum vector, i.e. $\int dp_x dp_y$ and the substitution for this is given by
$$ \xi = |\vec{p}| + |\vec{p} + \vec{q}| , \, \, \, \eta = |\vec{p}| - |\vec{p} + \vec{q}|$$...
- 1
1
vote
1
answer
27
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Flux through a paraboloid in the first quadrant
The question is absolutely easy but I am unsure where I go wrong haha. I am given a vector field $\mathbf{u} = (y,z,x)$ and I need to find the ourward flux $\iint\limits_{M} \textbf{u} \cdot d\textbf{...
- 617
0
votes
1
answer
68
views
What have I done wrong in this double integral?
The exercise I was solving asks to calculate $$\iint_{D} ydxdy$$ where $D$ is limited by $x=0, x^2+y^2=1, y^2=2x, y\ge0$.
What I have done is to use the graphs of $x^2+y^2=1$ and $y^2=2x$ in order to ...
- 185
0
votes
0
answers
56
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Proof that $\int_{C} f(z)dz =\int_{C_1} f(z)dz +\int_{C_2} f(z)dz + \cdots \int_{C_n} f(z)dz$
Let $C=C_1 \cup C_2 \cup \cdots \cup C_n$, where the individual pieces $C_i$ are of class $C^1$, i want to prove that
$$\int_{C} f(z)dz =\int_{C_1} f(z)dz +\int_{C_2} f(z)dz + \cdots \int_{C_n} f(z)dz$...
- 1,810
1
vote
0
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21
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How do I set boundaries in a multiple definite integral equation?
I've been trying to solve these types of problems for a while now and still can't figure out how to set boundaries for each integral in a multiple integral equation. One of the questions is as follows:...
- 11
1
vote
0
answers
46
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A mistake in Munkres' Analysis on Manifolds about proving if $D$ has measure zero in $\mathbf{R}^{n}$, then $\int_{Q}f$ exists
A mistake in Munkres' Analysis on Manifolds about proving if $D$ has measure zero in $\mathbf{R}^{n}$, then $\int_{Q}f$ exists
In Munkres' Analysis on Manifolds, page 94 Theorem 11.2 it states: Divide ...
- 174
0
votes
1
answer
40
views
Regions whose area cannot be measure in the sense of Riemann
Consider a region $D$ of the plane. Usually, its area is defined as
$$
\iint_D 1
$$
Would it be possible that this integral does not exists, and thus $D$ has a non-measurable area?
- 284
-4
votes
1
answer
80
views
Prove or disprove that $\iint\cos(x+y)$ of the function over region $D= \{(x,y):0\le x\le 1, 0\le y\le1\}$ is strictly between $0$ and $1$. [closed]
The question is to do this without integrating the function, since we have not learned double integration techniques. Is there a way to do this with the Riemann sums? Thanks
0
votes
1
answer
62
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If $y^2 - (x^2)y -2x=0, x, y >0$ and $\int \frac{(y-x^2)}{(x^2 +y)(y^2 +x)} dx=f(y) +C$ , where $c$ is a constant, find the value of $f'(y)$ at $x=1$. [closed]
I need help with this question:what I have done so far
$y^2 - (x^2)y -2x=0 \\
y-x^2 =\frac{2x}{y}...(1)$.
Substituting $1$ in integration I get $\int \frac{(2x)}{y(x^2 +y)(y^2 +x)} dx$. After this I ...
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
1
vote
2
answers
59
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Average distance from a point on a circle to the y-axis.
This is a simple question, but I must be making some mistakes as I don't seem to get the answer in the book.
Question: Determine the average distance from a point on $x^2+y^2 = 9$ to the $y$-axis.
My ...
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