All Questions
Tagged with bounds-of-integration definite-integrals
30
questions
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Prove/disprove upper bound and lower bound of the Integral
Hey I need to Prove or disprove this sentence:
$$
\frac{4}{9}(e-1) \leq \int_0^1 \frac{e^x}{(1+x)(2-x)} \, dx \leq \frac{1}{2}(e-1)
$$
using the infimum and supremum method for integrals, where m and ...
- 1
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75
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How is this property of definite integral derived?
The property:
$$
\int_a^b f(x) \, dx=\int_a^b f(a+b-x) \, dx
$$
Derivation given in my textbook:
Let $t = a+b-x$. Then $dt = -d x$. When $x=a, t=b$ and when $x=b, t=a$. Therefore,
$$
\begin{aligned}
\...
- 33
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votes
1
answer
46
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For which lower bound of integration $a$ does a definite integral of $f(x)$ from $a$ to $x$ equal its antiderivative $F(x)$ with $C=0$?
For an arbitrary antidifferentiable function $f(x)$, my goal is to construct a definite integral of $f(x)$:
$$
\int_a^x f(t) dt
$$
which is equal to one of the infinitely-many antiderivatives of $f(x)$...
- 43
0
votes
1
answer
75
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triple integral pyramid bounds
I am still confused about how to set up bounds for double and triple integrals. My task is to set up bounds for a function that is a pyramid with edge coordinates $(5,+-5, 0)$, $(-5,+-5,0)$, $(0,0,4)$....
- 1
1
vote
0
answers
106
views
Changing the Order of Integration in a Triple Integral
I'm currently studying for my multivariable calculus exam and I've come across a problem that I can't seem to solve. I have a triple integral with the order of integration $dz \, dy \, dx$ and I need ...
- 11
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votes
2
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119
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An upper bound for an integral
I saw many references using the following estimate but I couldn't prove it.
Given $T>0$ and $0 < b \leq \frac{1}{2}$, exist $C(b)$ constant that depends only on $b$ such that
\begin{equation}
\...
- 119
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2
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90
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Why don't the bounds in this definite integral change?
The question
This is probably a very basic question but I'm having a brain lapse and don't know why they didn't change the definite integral bounds from ($0 \rightarrow4$) to ($4 \rightarrow20$). I ...
- 5
1
vote
1
answer
168
views
Question on the bounds of definite integration during a substitution
Apologies if this question is rather elementary. I seem to still misunderstand a few things about how bounds change during substitutions still.
I was taught in calc II that to perform a substitution, ...
- 3,515
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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
2
votes
2
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226
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Volume bounded between sphere and three planes
I found a question in my homework that I have been trying to solve for days with minimal progress. We're given a sphere of form $x^2+y^2+z^2=9$ and three planes, $x=1,y=1,z=1$
The sphere in question:
...
- 33
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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54
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How can I find $\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx$ where $1≤x^2+y^2≤3, 0≤z≤3$?
Compute $$\iiint\frac{xz}{1+x^2+y^2}\,dz\,dy\,dx,$$ where $1≤x^2+y^2≤3, 0≤z≤3$.
I've tried it. But I'm only confused with $\theta$. I think it should be $0$ to $2\pi$, but that'll make the whole ...
5
votes
1
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409
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How can you simplify/verify this solution for $\int\limits_0^{.25991…} Q^{-1}(x,x,x)dx?$
As I do not know the complex behavior of this function, it would be even harder to integrate past the real domain. The upper bound for the domain is a constant I will denote β.
$${{Q_2}=\int_0^βQ^{-1}(...
- 12.5k
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113
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Quick question about integration of a piecewise defined function
I have the following "quick" question about the piecewise function integration:
Say, I have to find $\int\limits_0^{1}f(x)d x$, with $f(x)$ being piecewisely defined on $\mathbb{R}$ as ...
- 1,170
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2
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634
views
Evaluate $\iiint_{R} (2x+y) \,dx \,dy \,dz$
Evaluate $$\iiint _{R} (2x+y) \,dx \,dy \,dz\,,$$ where $R$ is the region bounded by the cylinder $z = 4 - x^{2} $ and the planes $x = 0$, $y = 0$, $y = 2$ and $z = 0$.
How do I extract the limits ...
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