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.
75,365 questions
3
votes
0
answers
47
views
Non-elementary integral which becomes elementary through a geometric argument
I occasionally wonder about the following question when I teach the washer and shell methods in Calculus II.
Is there a real invertible function $f:[a,b]\to[c,d]$ with both $f$ and $f^{-1}$ being ...
- 559
3
votes
0
answers
37
views
Compute the integral $\int_{-\pi}^{\pi} \frac{\cos^4 x + \frac{x \sin x}{1 + \cos^2 x}}{1 + e^{-x} } dx$ [closed]
Compute the integral
$$\int_{-\pi}^{\pi} \frac{\cos^4 x + \frac{x \sin x}{1 + \cos^2 x}}{1 + e^{-x} } dx$$
This is one of exercises in my real analysis textbook, and I am frustrated by it since I ...
- 41
0
votes
0
answers
41
views
Is the expected length of a line in a disk proportional to the radius?
Suppose we have two i.i.d random variables $Q_1 = (x_1, y_1)$ and $Q_2= (x_2, y_2)$ that uniformly randomly sample points from a disk of radius $R$, $D_R$. Denote the new random variable $\ell = ||Q_1 ...
-2
votes
0
answers
16
views
How to find k value? [closed]
v(dx/dW) = k((1-X)/(1+ 2 X)). [[ 2.43 x 10^(-3)W(1+2X)+1]^(1/2)]
Given values are v=6.266,W=from 0 to 72.66, x= from 0 to 0.98, find the k value
0
votes
0
answers
27
views
Getting area between a lens and an ellipse to approximate $\pi$
I have here a desmos graph. You see a tiny bit of space in between the lens and the ellipse, some of which I have approximated using a triangle. Now, I need to calculate the rest of the space between ...
- 161
1
vote
0
answers
62
views
Evaluating $\int e^{x^4} (\cos(x)-x^3) dx$
I am trying to evaluate the following integral:
$$\int e^{x^4} (\cos(x)-x^3) dx$$
Since $\int e^{x^4} (\cos(x)-x^3) dx=\int e^{x^4}\cos(x)dx-\int x^3 e^{x^4} dx$ and clearly $\int x^3 e^{x^4}dx=\...
- 4,180
-1
votes
1
answer
44
views
How to express this integral using Meijer G-function or special functions? [closed]
I am trying to evaluate the following integral:
$$
\int_T^{\infty}{\frac{e^{-\beta w}w^{\alpha}}{w^{\alpha}+C}\mathrm{d}w},
$$
where $$C,T,\alpha,\beta>0.$$ The integral involves an exponential ...
- 1
2
votes
1
answer
42
views
Double Integral of $3y$ over a Region Defined by a Circle
I am trying to evaluate the following double integral:
$$ \int_0^1 \int_{-\sqrt{1 - y^2}}^{\sqrt{1 - y^2}} 3y \, dx \, dy $$
Can anyone help me with the solution or provide insights into the process?
...
- 21
1
vote
1
answer
18
views
Transformation Formula for distinct Dimensions?
I am currently dealing with an integral for which I am not sure whether I can apply a transformation formula. The situation is essentially the following. Let $T\in\operatorname{Hom}(\mathbb R^2, \...
- 572
0
votes
0
answers
19
views
$E(a, b) \pm F(a, b)$ Elliptic integral identity
I found this addition theorem of $F$ and $E$ on Wikipedia, but there’s no addition theorem that combines $F$ and $E$ together specifically
$$F\bigl[\arctan(x),k\bigr] + F\bigl[\arctan(y),k\bigr] = F\...
- 1,028
0
votes
0
answers
21
views
How to Derive the Exact Normal Force for a Wire Sliding Around a Tube Using the Capstan Equation?
I’ve been working on a problem that combines friction mechanics and abrasive wear, and I’ve hit a point where I could use some help. The issue revolves around calculating the exact normal force for a ...
1
vote
0
answers
22
views
Verifying a spectral integral equality
I'm stucked when reading Chemin's Mathematical Geophysics An introduction to rotating fluids and the Navier-Stokes equations. On page 38 he applied spectral theorem to Stokes operator $\mathcal{A}$, ...
- 65
1
vote
1
answer
45
views
Work required to fill spherical tank at height
Suppose we have a spherical water tank $24$ feet in diameter, sitting atop a $60$ foot tower. The tank is filled by a hose attached to the bottom of the sphere. How long does it take a $1.5$ ...
- 123
0
votes
1
answer
95
views
+50
Find the area between two graphs can't get to $4\ln{2} - 2\ln{3}+1$
Find the area between two graphs, $f(x) = \frac{2}{x}$ and $ g(x)=x-1$ on the interval $1 \le x \le 3$.
Integrals:
$$Area = \int_{1}^{3} (g(x) - f(x)) \, dx = \int_{1}^{3} (x-1) \, dx - \int_{1}^{3} (...
- 297
0
votes
0
answers
45
views
Integral $\int_x^{2\pi}\tan(u)\, du$ can be computed for which $x$?
Computing
$$
\int_x^{2\pi}\tan(u)\, du
$$
one can use that
$$
\frac{d}{dx}(-\ln(\cos x))=\tan(x)
$$
so that
$$
\int_x^{2\pi}\tan(u)\, du = \ln(\cos(x))
$$
But I am wondering for which $x$ this is true,...
- 551
0
votes
1
answer
38
views
Find the shaded area in figure two lines
Find the area of the shaded region:
The limitation I am able to find from the figure: $\left[-2, 0\right]$ and $\left[0, 1\right]$.
Integrals:
$$\int_{-2}^{0} (x + 2) \, dx + \int_{0}^{1} (2e^{-x}) \,...
- 297
8
votes
2
answers
127
views
Asymptotes of an integral
Given the function
$$f(x,y)=\int_0^{\infty } \frac{1}{\sqrt{u}} e^{-\frac{x^2}{u}} e^ {-\frac{(y-u)^2}{2}} \, du$$
for $x>0$, I am wondering if it is possible to find an explicit expression for it, ...
- 305
1
vote
1
answer
41
views
Is there any solution $u\left(t,x\right)$ that satisfies the following system of equations?
Is there any solution $u\left(t,x\right)$ that satisfies the following equation (sorry for the “oddly specific formula”)?
$$\begin{cases}
\vartheta\left(\mu-x\right)=\frac{a}{b}\sigma-\frac{\sigma}{b}\...
- 1,446
0
votes
0
answers
57
views
Is there another way to prove this Integral equality?
There is a problem from my Calculus textbook that:
Prove that: For $f^{(n+1)}(x)$ integrable on [a,b], $n\ge0$, $$f(b)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(b-a)^{k}+{1\over n!}\int_{a}^bf^{(n+1)}(t)(b-t)^...
- 115
3
votes
1
answer
60
views
Show that $I(\gamma) = \int_0^{\infty} x^2 e^{-\gamma x} \, dx$ is continuous
Let $I(\gamma) = \int_0^{\infty} x^2 e^{-\gamma x} \, dx$. Show that $I$ is continuous for $\gamma > 0$.
My approach: let $\gamma_n$ be a decreasing sequence such that $\gamma_n \to \gamma$. We ...
- 171
1
vote
4
answers
225
views
Integrating $\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}} \,dx$
$$\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}}
\,dx$$
Now I have seen a similar integral here,
I am unable to deal with the $x^{\frac12}$ term in the denominator.
Here's one idea,
$$\int_0^1x^...
- 5,650
1
vote
1
answer
55
views
Showing that this limit of an integral exists
I am struggling with the following exercise:
Let $f(x) = \frac{\sin(x)}{x}$, which is continuous on $\mathbb{R}$.
Show that $\lim_{T \to \infty} \int_{0}^{T} f(x) \, dx$ exists (Hint: Use Chasles' ...
- 395
1
vote
1
answer
43
views
Trying to prove the equivalence of the two definitions for the Gini index. Where am I making a mistake?
The Gini index is a quantity that's often used to characterize income or wealth disparity. It can be defined by the Lorentz curve or a scaled sum of pair distances. Here I'm trying to prove that these ...
- 303
1
vote
0
answers
68
views
Defining the area enclosed by a Jordan curve.
I specifically made an account just to ask this question and it is thus my first post. Please excuse any errors or faux-pas in posing this question.
Summary:
Let $\gamma$ be a Jordan curve (simple ...
- 11
1
vote
1
answer
37
views
Using Disc Method getting wrong Volume
Find the Volume, using: $$V = \pi \int_{a}^{b} (f(x))^2 dx$$
$$y = \frac{x}{2} + 1$$
The limitation I am able to find from the figure,$\left[1, 5\right]$.
Integral:
$V = \pi \int_{1}^{5} (\frac{x}{2} ...
- 297
0
votes
1
answer
93
views
Fubini's Theorem: $\int_{2}^x (\int_2^t \frac{du}{log(u)})\frac{1}{\sqrt{t}}dt = \int_{2}^x \int_u^x \frac{1}{\sqrt{t}\cdot log(u)}du dt=...$
I have
\begin{align}
\int_{2}^{x}\left[\int_{2}^{t}
\frac{{\rm d}u}{\log\left(u\right)}\right]\frac{1}{\sqrt{t}}\,{\rm d}t & =
\int_{2}^{x}\int_{u}^{x}
\frac{1}{\sqrt{t}\,\log\left(u\right)}\,{\...
- 67
0
votes
0
answers
43
views
Condition for an elliptic integral to be solvable
When we are faced with integrals like this, how would one know if the integral is solvable or unsolvable
$$f(x) = \int \sqrt{ P(t)} \,dt$$
By solvable I mean the integral can be broken down into an ...
- 1,028
3
votes
2
answers
140
views
can $\int \sqrt{1 + \frac{1}{x^4}}dx$ be expressed in terms of elementary functions?
I'm trying to evaluate this integral:
$$\int \sqrt{1 + \frac{1}{x^4}}dx$$
After a little substitution, the integral can be shown to be equivalent to
$$\int \sqrt{\cosh(2\alpha)}\,d\alpha,$$
which can ...
- 35
0
votes
1
answer
33
views
A change of variables for a complex function
Consider the function $\phi(z):=\tanh((\pi/4) z)$, which maps $\mathbb{R} +i$ bijectively to $\{e^{i\theta} ; \theta \in (0,\pi)\}$, i.e. the upper half of the Torus. I am working through a paper ...
- 127
0
votes
1
answer
60
views
Problem with integration by parts for a cumulative probability and probability density functions
In Keller (1974) paper titled “Optimum Checking Schedules for Systems Subject to Random Failure”, the author derives expected cost function for linear loss function case as follows:
The initial ...
-4
votes
0
answers
54
views
How they took x=y if there already exist relation between x and y [closed]
solutionLet f be a one-to-one continuous function such that f(2) = 3 and f(5) = 7 Given $$\int_2^5 f(x)\, dx = 17$$ then find the value of $$\int_3^7 f ^{- 1} * (x) dx$$
In the solution in third line ...
3
votes
1
answer
261
views
Integral inequality proof
Problem: Let f(x) be a convex function and integrable on [a, b]. Prove that
$$ \frac{1}{b-a} \int_{a}^b f(x)dx \le \frac{f(a) + f(b)}{2} $$
Here my try: Since f(x) is a convex function so it is ...
- 115
7
votes
2
answers
212
views
Computing a definite integral knowing its equation
Here the problem: Let f(x) continuous on [-1,1] such that $$f(x) = \sqrt{1-x^2} + x^2f(x^2) (*)$$ Calculate $$ \int_{-1}^1 f(x)dx $$
EDIT: Actually, I found this problem from a calculus 1 textbook ...
- 115
0
votes
1
answer
105
views
How to prove that the area bounded by the curve: $x^4+y^4+2 x^2 y^2 -m x^2-(1-m) y^2=0$ is a fixed value? (when 0＜m＜1) [closed]
I tried using polar coordinate definite integral, it's easy enough, but is there any easier or more interesting method to prove that?
The origin of this problem:
Through a point inside the circle, ...
- 27
1
vote
1
answer
46
views
Integration wrt increasing functions and measures
Suppose that $\mu_n$ and $\mu$ are measures on $(X, \Omega)$ such that $\mu_n$ increases to $\mu$. If $f_n$ and $f$ are nonnegative measurable functions such that $f_n$ increases to $f$, then
$\int_X ...
- 13
3
votes
1
answer
59
views
What makes the method of directly multiplying differential elements in a change of variables for double integrals incorrect? [duplicate]
When applying a change of variables in double integrals, the relation containing the Jacobian is as follows:
$\iint_D f(x, y) \, dx \, dy = \iint_R f(x(u, v), y(u, v)) \, |J| \, du \, dv $
where is ...
- 165
0
votes
0
answers
36
views
Using the weierstrass substitution for integration and limits [closed]
I was doing a question where the limits in x where above and below pi and I had to split the integral at pi, as tan(pi/2) is asymptotic. I want to know when to do this - must I split when x=npi, for ...
- 9
-1
votes
0
answers
21
views
change limit and integral as function is uniformly continuously [closed]
Can someone help me?
I come across this argument but I still struggle to proof ist:
It says I can switch the limit and the Integral if the funktion is uniformly continuous. I tried to proof it using ...
- 1
3
votes
0
answers
33
views
Link between integration of differential forms and complex measures
Suppose that $M$ is a compact smooth manifold of dimension $n$. Let $\omega$ be a $n$-form on $M$ (complex-valued in the sense that $\omega_x$ is a complexified alternated $n$-covector for each $x\in ...
- 891
1
vote
2
answers
59
views
Confusion with calculating a limit $F(x):=\int_{x}^{2x}e^{-2t}t^{-1}dt$
The question is given in the following way:
Define $F:(0,\infty)\rightarrow \mathbf{R}$ by $$F(x):=\int_{x}^{2x}e^{-2t}t^{-1}dt$$ Determine whether or not $\lim_{x\rightarrow 0^+}F(x)$ exists.
In ...
- 164
2
votes
0
answers
39
views
Proving $\int^{\infty}_{-\infty} \frac{p(x-\mu)}{0.5p(x-c) + 0.5p(x+c)} p(x) d x \leq 1$ for symmetric and monotone Lebesgue density $p$.
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and that $p(x) \...
- 21
1
vote
0
answers
44
views
Evaluating $\int \frac{1}{\sqrt{2(y-a \ln(y+a)+c)}}dy$
I am trying to solve this nonlinear ODE equation:
$y''=\frac{y}{y+a}$
I tried to $v=y'$ and it gives me
$v=\pm\sqrt{2(y-a \ln(y+a)+C_0)}=\frac{dy}{dx}$
which requires me to solve
$x=\pm \int{\frac{1}{\...
- 11
1
vote
1
answer
53
views
Where did I go wrong in this splitted curve Integral problem
Find the area between the curve $y = 2x^3 - x^2 - 1$ and the $x-$axis, in range limitation $-2 \le x \le 2$ i.e., $[-2, 2]$.
$$\int_{-2}^{2} \left(2x^3 - x^2 - 1\right) \, dx$$
\begin{align}\text{Area}...
- 297
3
votes
3
answers
270
views
Evaluating $\int_0^1 \frac{\tan^{-1}(x)\ln^2(x)}{1+x}\,dx$
$$I=\int_0^1 \frac{\tan^{-1}(x)\ln^2(x)}{1+x}\,dx$$
I have not been able to find any similar integrals to refer and solve this one.
$$\int_0^1 \frac{\tan^{-1}(x)\ln^2(x)}{1+x}\,dx\underset{ibp}=-\...
- 5,650
1
vote
3
answers
111
views
Area of the shaded region $\frac{41}{6} + \frac{2}{3}$ but is $\frac{59}{6}$
Find the area of the shaded region using primitive function.
$$f(x) = x^2 + 5x + 4$$
$X_1 = 1, X_2 = -2$
The limitation I am able to find from the figure,$\left[-2, 1\right]$.
Integral:
$$\int_{-2}^{...
- 297
-1
votes
0
answers
83
views
How to calculate $\int_{-\infty}^z{\frac{\mathrm{arctan} \left( z-x \right)}{1+x^2}dx} $ [closed]
Problem Description:
Let $X$ and $Y$ be independent and identically distributed (i.i.d.) standard Cauchy random variables. We are interested in finding the distribution of $Z = X + Y$. The cumulative ...
- 75
1
vote
1
answer
75
views
Mellin transform of the power of the reciprocal gamma function
Let $$f(z)=\frac{1}{\Gamma^n(z)}\text{, $n>0$: integer, and $z>0$}.$$ How to obtain the Mellin transform of $f(z)$? I couldn't find it in any table of Mellin transforms. thanks in advance.
My ...
- 197
2
votes
4
answers
969
views
Area of shaded curve integral negative thus incorrect
Find the area of the shaded figure, integral.
Integral:
$$\int_{a}^{b} (x^2-4) \, dx$$
Limitations:
$y=x^2-4$
$x^2-4=0$
$x^2=4$
$x = \pm \sqrt{4} = 2$
The limitation I am able to find from the figure,...
- 297
0
votes
0
answers
37
views
How to show this inequality that the norm of two integrals is no bigger than the integral of the norm? [closed]
$\newcommand{\on}[1]{\operatorname{#1}}$
If $\,\on{f}$ and $\on{g}$ are two integrable functions on probability space
$\left(\Omega, \mathcal{F}, P\right)$,
$$
\mbox{how to show that}\
\sqrt{\,\mathbb{...
- 331
1
vote
0
answers
124
views
An Awkward Piecewise Integral from Integration Bee Austria Fall 2024!??
Compute the integral:
$$\int_1^{\infty}\frac{1}{\left\lfloor \sum_{n=1}^{\left\lfloor x \right\rfloor}\frac{\left\lfloor x \right\rfloor(-1)^{\left\lfloor x+1 \right\rfloor}}{n^{\left\lfloor x \right\...
- 1,531