All Questions
Tagged with bounds-of-integration real-analysis
7
questions
0
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0
answers
28
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Bounding an exponential row
Let $0<c<1$. I need to bound
$$
\sum_{i=1}^n \frac{c^{n-i}}{i}\leq C n^{-?}
$$
for some constant $C>0$. Does anyone know how to optimal bound this sum?
Thank you very much for any suggestions....
- 155
1
vote
1
answer
78
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Evaluate $\int \int \int_B x^2+y^2 \, dxdydz$
I'd like to evaluate $\int \int \int_B x^2+y^2 \, dxdydz$ where $B$ is the area enclosed by $x^2+y^2=2z$ and $z=2$ but I'm not sure about the bounds. I've thought something like this...
$$\int_0^2 \...
- 1,604
3
votes
3
answers
889
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$\iiint_M (x+y+z)\,dx\,dy\,dz$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$
$$\iiint_M (x+y+z)\,dx\,dy\,dz$$ over $M=\{(x,y,z)\in\mathbb{R^3}: 0≤z≤(x^2+y^2)^2≤81\}$. How would I express this with the correct bounds? Once I have the bounds I can continue on my own but I need ...
- 487
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45
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Conditions on a rate of change of a continuous function to be bounded
Suppose $f(s)$ is continuous on $[0,\infty)$ and $\lim_{s\to \infty} f(s) =1$. How fast should it decrease to $1$ so that $$F(t)=\int_0^t f(s)\sin(s)ds$$ to be bounded? In what cases it is ...
0
votes
2
answers
149
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Does the existence of the integral $\int_0^\infty f(x)dx$ imply that f(x) is bounded on $[0,\infty)$ when f(x) is continuous in this same interval?
Does the existence of the integral $\int_0^\infty f(x)dx$ imply that f(x) is bounded on $[0,\infty)$ when f(x) is continuous in this same interval ?
edit
I'm trying to use Cauchy without knowing ...
- 29
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votes
1
answer
65
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Differentiation Under the Integral Sign w Variable Substitution
Let $f\in C^1(B_\rho(\xi))$, $\xi\in\mathbb{R}^n$ and $\rho>0$. I wish to show
$$ \frac{d}{d\rho} \int_{B_\rho(\xi)} f(x)\ dx = \frac{1}{\rho} \int_{B_\rho(\xi)} nf(x) + x\cdot Df(x)\ dx $$
I'm ...
user512116
1
vote
1
answer
63
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A confusion about find the boundary of a set $E = D \times E_x$, where $D \subseteq \mathbb{R}^n $ and $E \subseteq \mathbb{R}^1 $
In the book of Mathematical Analysis II by Zorich, at page 132, it is given that
However, in the proof of the Remark, I couldn't understand how does the author conclude that $\partial E$ is the ...
- 7,337