Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
73,994
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The volume of two vertically stacked, diagonally halved unit cubes
How do I express the volume of the painted area using multiple integrals? The answer should be $1$, because the volume is $0.5 \times 2 \times 1 \times 1=1$.
- 11
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0
answers
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A way to calculate volume of a zip-loc bag using only mathematical methods
The Zip-loc challenge is pretty old, but I was doing it today to test my knowledge of chemistry. The way I calculated the volume of a ziplock bag was to fill to with water and find the calculate how ...
user531621
2
votes
1
answer
189
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How do you integrate $t\coth(x t)/\sqrt{1-t^2}$ from $t = 0$ to $1$?
How do you do this integral:
$$A=\int\limits_{0}^{1}\frac{t}{\sqrt{1-t^2}}\coth{(xt)}\mathrm{d}t$$
for a positive parameter x.
- 103
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votes
1
answer
77
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How to double integrate
Graph
The $z$ axis is acceleration and the $x$ and $y$ axis is time. The painted area is $\frac{1}{2}at^2$, and I don't understand why the diagonal line is velocity instead of an area made by the ...
- 11
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2
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Sketch and find the volume of the solid in the first octant bounded by the coordinate planes, plane x+y=4 and surface z=root(4-x)
I understand that this can be done with triple integrals, but my class has yet to be taught those and we will be assessed on our ability to perform a question similar to this one with the principles ...
- 29
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1
answer
3k
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Proving error bound for Simpson's rule
The Simpson's rule can be stated as follows:
$$\int\limits_{x_0}^{x_2}f(x)dx\approx \frac{h}3\left[f(x_0)+4f(x_1)+f(x_2)\right]$$
The way I'm trying to find the error bound for the Simpson's rule is ...
- 9,648
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1
answer
907
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find the simple closed curve of $F(x,y) = (y^3-6y)i + (6x-x^3)j$ using Green's Theorem which will have the largest positive value
$F(x,y) = (y^3-6y)i + (6x-x^3)j$
a. Using Green's Theorem, find the simple closed curve C for which the integral
$ ∳F \cdot dr $ (with positive orientation) will have the largest positive value.
b....
- 29
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votes
2
answers
115
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Consider the vector field $F=-c \frac{x\mathbf{i}+y\mathbf{j}}{x^2+y^2}$.
$$\mathbf{F}={-c}\frac{x\mathbf{i}+y\mathbf{j}}{x^2+y^2}$$
(vector field was rewritten here to make it easier to see)
Consider the vector field above and using $c=1$, find by direct calculation the ...
- 29
6
votes
2
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419
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Evaluate $\int x^x \ln x\, dx$
The integral $$\int x^x \ln x\, dx= ?$$
I know of the integral $\int x^x dx$ can be further simplified as $\int e^{x\ln x} dx$. And this requires identity to simplify. What about the product in the ...
- 391
2
votes
2
answers
65
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Puzzled with a double integration
Double integration of $f(x,y) = \frac{(x-y)}{(x+y)^3}$ for which $x$ ranges from $0$ to $1$ and
$y$ ranges from $0$ to $1$. It comes out to be $\frac{1}{2}$.
But suppose that for $(a,b)$ the value of $...
5
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2
answers
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Graphically, why is $\int_{0}^{1} \frac{1}{x} dx$ divergent but $\int_{0}^{1} \frac{1}{x^{0.999}} dx$ convergent?
When the power of $x$ is less than 1, it seems that the improper integral converges. I understand the math, but I don't understand how the graphs of the two cases $\frac{1}{x}$ and $\frac{1}{x^{0.999}}...
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2
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80
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Integrate $\frac{dx}{x^{2n}+1}$ for $n \in \mathbb{N}$ from $-\infty$ to $\infty$ [duplicate]
How can one calculate
$$
\int_{- \infty}^{\infty} \frac{dx}{x^{2n} + 1}, \;n \in \mathbb{N}
$$
without using complex plane and Residue theorem?
- 191
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votes
1
answer
174
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closed form for $\int_{1/4}^{3/4} x^n(1-x)^n \, dx$
The integrand being a polynomial, I used the binomial formula to separate the monomials:
$$\int_{\frac{1}{4}}^{\frac{3}{4}} x^n(1-x)^n \, dx = \sum_{k = 0}^{n}{ n \choose k}(-1)^{k}\int_{\frac{1}{4}}...
- 2,425
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5
answers
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Quick way of solving the contour integral $\oint \frac{1}{1+z^5} dz$
Consider the contour integral in the complex plane:
$$\oint \frac{1}{1+z^5} dz$$
Here the contour is a circle with radius $3$ with centre in the origin. If we look at the poles, they need to satisfy $...
- 839
2
votes
0
answers
48
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Extension of a integration by parts formula for a linear operator on $C_0(\mathbb R)$
Let
$(\mathcal D(A),A)$ be a linear operator on $C_0(\mathbb R)$ (the space of continuous functions vanishing at infinity equipped with the supremum norm $\left\|\;\cdot\;\right\|_\infty$) such that $...
- 13.4k
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0
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Mass of $ A := \{ x \in R^2\ | x<y , y<2, y>1/x \}$
How does one calculate the Lebesgue Measure of the above set?
I tried the following:
$ \lambda^2(A) = \int_{R^2} 1_A d(\lambda^2(x,y)) = \int_R\int_x^2 1_{ \{1/x < y\} } d\lambda(y)d\lambda(x)$
...
- 81
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1
answer
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Complex Contour Integrals Answer check [closed]
can someone check my answers for the following question:
Evaluate the integrals $\ \int{z^2 dz} $ and $\ \int{|z|^2} dz $ along the following paths
a) line from 1 to i
b) quarter of the unit ...
- 5
1
vote
1
answer
162
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$\int_0^t e^{sA}\cos(\omega s)ds$ with $A$ matrix
Let $A$ be a singular square matrix and $\omega,t\in\mathbb{R}^{*+}$. How to compute the following integral?
$$I = \int_0^t e^{sA}\cos(\omega s)\,\mathrm{d}s$$
Since I am looking for a numerical ...
- 3,514
1
vote
0
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42
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Taylor series for with an integral
I was trying to analyze a large amplitude problem and I got stuck at an equation like this.
$$\int_0^T dt=\sqrt{\frac{l}{2g}}\int_{\theta_{\text{max}}}^{ \theta_0} \frac{d \theta }{ \sqrt{\cos\theta -...
- 19
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votes
1
answer
53
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$\int _ { 0 } ^ { L } \left\{ \left( \int _ { 0 } ^ { x } f ( y ) d y \right) \left( \int _ { 0 } ^ { x } g ( z ) d z \right) \right\} d x$
As in the title,
What I wanna do is
$\int _ { 0 } ^ { L } \left\{ \left( \int _ { 0 } ^ { x } f ( y ) d y \right) \left( \int _ { 0 } ^ { x } g ( z ) d z \right) \right\} d x$
Here, functions f and g ...
0
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0
answers
223
views
$\int_{100}^{200} \frac{1}{ 8 + b\sqrt{1-\frac{(x-100)^2}{100^2} } + b\sqrt{1-\frac{(x-200)^2}{100^2} } } dx$
$\int_{100}^{200} \frac{1}{ 8 + b\sqrt{1-\frac{(x-100)^2}{100^2} } + b\sqrt{1-\frac{(x-200)^2}{100^2} } } dx$
I tried this in Mathematica but got the same thing as solution. By considering "b" = ...
- 1
-1
votes
4
answers
426
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Inverse trig function integration by parts.
$$\int \tan^{-1}{2y}dy$$
if I choose $u = \tan^{-1}{2y}$ then $du = \frac{2}{1 + (2y)^2} dy$
and
$dv = dy$ and $v = y$
But I have a more complicated du. What else can I do?
- 5,714
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votes
2
answers
136
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Integration by parts questions. Work check.
I have a couple of problems that I'm trying to work through. I'm a tad stuck on 2. Here is what I have?
$\int t \cdot e^{-3t} dt$
so let's say:
$$u = t \quad \text{and} \quad du = dt$$
$$...
- 5,714
4
votes
2
answers
93
views
Find $\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$
$$I:=\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$$
Here the region of integration is the triangle with vertices $(0,0),(0,1)$ and $(1,1)$ and given as a type-1 region. We can convert it into a type-2 ...
- 1,196
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votes
3
answers
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Integral $\int\frac{1}{1+x^3}dx$
Calculate$$\int\frac{1}{1+x^3}dx$$
After calculating the partial fractions I got:
$$\frac{1}{3}\int\frac{1}{x+1}dx+\frac{1}{3}\int\frac{2-x}{x^2-x+1}dx=\frac{1}{3}\ln(x+1)+\frac{1}{3}\int\frac{2-x}{...
- 599
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votes
4
answers
1k
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Bringing infinite limit inside integral
I want to find $$\lim_{R \to \infty} \int_{0}^{\pi} e^{-R\sin (t)}dt.$$ Since $\sin(t)$ is nonnegative on $[0,\pi]$ the integrand vanishes as $R \rightarrow \infty$. So I want to bring the limit under ...
- 412
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1
answer
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Integral $\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2}dx$
I have difficulties with calculating the following integral:
$$I(a)=\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2} \mathrm dx~~~~~~~,\text{where } a>1$$
For the case with $a=1$ the ...
- 129
3
votes
1
answer
52
views
Evaluate the sign of an integration
Let $P_k(x)$ to be the first $k+1$ terms of the Taylor expansion of $\cos(x)$, that is
$$P_k(x) = \sum_{l = 0}^k (-1)^l x^{2l}/(2l)!.$$
For $\alpha>0$ and $\alpha\notin \mathbb{Z}$, I want to ...
- 335
0
votes
1
answer
127
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Solving a real integral in the complex plane
$\int_0^\infty \frac{\cos(x)}{x^2+1}dx$
Singularities: $x_{1,2}=\pm i$
We want to integrate over the upper half of a circle on the complex plane. So we only consider $x_1=+i$. We can use the residue ...
- 887
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votes
1
answer
36
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What are the Integration-by-parts steps needed to get from Equation 1 to 2?
I need help justifying the jump from the expression on the left side of the equal sign, to the right.
\begin{array},\int_0^L \mathrm{d}x \ \frac{\partial y}{\partial t} \frac{\partial}{\partial x} ...
18
votes
2
answers
635
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An intriguing pattern in Ramanujan's theory of elliptic functions that stops?
I. Define the ff integrals,
$$K(k)=K_2(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)=\int_0^{\pi/2}\frac{\cos\left(...
- 54.7k
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1
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one bound integral
so i have a formula for finding the center of mass of a body:
$\frac{1}{m}\int_V\vec r\,dm$
what does it mean when an integral has only one bound like this on the bottom?
0
votes
1
answer
25
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Why is $\int_{t_n}^{t_{n+1}} u'(s) ds - u'(t_n) = \int_{t_n}^{t_{n+1}}(t_{n+1}-s)u''(s) ds$
I don't understand why the last step in the following equations is true. Could someone explain this to me please? Don't think context is important here, but just in case it's from a proof of a bound ...
- 2,809
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votes
5
answers
823
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prove $\int_0^\infty \frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}{8}$ with real methods
Context: I looked up "complex residue" on google images, and saw this integral. I, being unfamiliar with the use of contour integration, decided to try proving the result without complex analysis. ...
- 17.2k
1
vote
1
answer
32
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calculating 2 constants in a function
$$M=\left\{f\in C[0,2\pi],\int_{0}^{2\pi}f(x)\sin x\,dx=\pi,\int_{0}^{2\pi}f(x)\sin2x\,dx=2\pi\right\} $$ $a,b\in \mathbb R, g\in M, g(x)=a\sin x+b\sin2x,x\in [0,2\pi]$
I've read on the answers ...
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1
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To calculate the final velocity, (v) of a moving body that had an initial velocity, (u)...
...and had been under constant acceleration, $a$ for a period of time, $t$ the following formula is used:
$$v = u + at$$
so the following terms are constant,initial vel.$u$ and the constant ...
0
votes
1
answer
48
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Simplification of an expression of a double integral
Is the following expression able to be simplified?
$$I = \int_0^{x/2} \int_0^s f(s,r)\,dr\,ds + \int_{x/2}^x \int_{2s - x}^s f(s,r)\,dr\,ds . $$
Here $f(s,r)=u(r,2s+x-r)$ so that we could also write
$...
- 1,523
3
votes
2
answers
218
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Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$
I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
- 1,859
1
vote
1
answer
58
views
If $g(x)$ be the inverse of $f(x)$ then prove that $2g''=3g^2$
Let $$f(x)=\int_0^x\frac{dt}{\sqrt{1+t^3}}$$ Prove that $$2g''=3g^2$$ given $g(x)$ is inverse of $f(x)$.
I tried of applying Newton-Leibnitz both sides but could not succeed as the variable is $x$ on ...
- 10k
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1
answer
75
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Express moments in terms of exponential series
Consider a random variable $X$ with cumulative distribution function
$$ F(x)= [G(x)]^{\alpha} $$
where $ G(x)$ is baseline distribution, and its survival function is $ R_{G}(x)=1-G(x)$.
Put $R_{G}(...
- 15
0
votes
1
answer
211
views
Fundamental Theorem of Calculus with Functions
Please excuse the weird title. I was unsure about how to summarize this.
So lets say we have this integral:
$ \int_{kx}^{cx} t^2 dt = f(x)$
If we wanted to apply $f(x)$ with $2x$ instead of $x$ ...
- 49
-1
votes
1
answer
38
views
how to make a curve that never goes down in y over time?
(for motion graphics) I need to take an audio clip's curve, and give it a rule: if you ever go down in the y axis, I need you to plateau instead. Attached is a sin curve that has that rule implemented,...
1
vote
1
answer
790
views
Integral of Reciprocal Functions: Why The Modulus Sign?
I've been taught that $\int\frac{1}{x}dx=\ln|x| + C$ rather than $\ln(x) + C$ without the modulus sign and been told that it is incorrect to not have the modulus sign. Why is that?
I've been ...
- 69
2
votes
2
answers
671
views
give 5 other equivalent iterated triple integrals
I am given the following integral:
$$\int_0^2\int_0^{y^3}\int_0^{y^2}f(x,y,z) dzdxdy $$
I was successfully able to rewrite this in its dzdydx, dxdzdy, and dxdydz forms, but I'm having a hard time ...
- 23
3
votes
3
answers
414
views
Uniform continuity implies existence of limit of integral
Let $f: (0,1) \rightarrow \mathbb{R} $ be uniformly continuous. Prove that $$ \lim_{\epsilon \to 0} \int^{1-\epsilon}_{\epsilon}\!\!f(t)dt \in \mathbb{R}$$ Any ideas?? $f$ can be extended to a ...
- 41
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0
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40
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If $Lf=bf'+\frac12σ^2f''$, $L^\ast$ is the $L^2$-adjoint of $L$, $ϱ$ is a solution of $L^\astϱ=0$ and $\mu=ϱ{\rm d}x$, is $L$ symmetric on $L^2(\mu)$?
Let $b\in C^1(\mathbb R)$, $\sigma\in C^2(\mathbb R)$, $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ and $$L^\ast g:=\frac12(\sigma^2g)''-(bg)'\;\;\;\text{for }g\in C^2(\mathbb R)....
- 13.4k
3
votes
1
answer
792
views
integration by substitution of multiple variables
I have an integral
\begin{equation}
\int_{\mathbb{R}^n}f(\mathbf{B}\mathbf{x})\mathrm{d}\mathbf{x}
\end{equation}
where $f: \mathbb{R}^m \rightarrow \mathbb{R}$ and $\mathbf{B}\in\mathbb{R}^{m\times ...
- 401
0
votes
2
answers
2k
views
How to find the power series expansion that converges to Fresnel integral?
Fresnel integral is $S(x)=\int_{0}^x{\sin(t^2)\,dt}$.
I'm trying to see how the power series expansion for the integral is found , I have to tried to use Taylor Series for expanding $\sin(t^2)$ but i ...
- 785
4
votes
0
answers
216
views
differential equation of the explicit RMS function
This is my first time posting on any math forum, let alone stackexchange, so I do hope I'm doing everything correct!
Some Background
I'm an engineer, and not a mathematician, although I do enjoy ...
0
votes
0
answers
71
views
how to find pdf $f_X(x)$ from joint pdf $f_{X,Y}(x,y)$
enter image description hereI have joint PDF of $X$ and $Y$. $X$ and $Y$ are dependent random variable. I know that $X$ and $Y$ have the same distribution. it is hard to integrate joint PDF. are there ...
