Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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15 views

Is this upper bound ok to use when bounding the error between the Riemann sum and its integral?

I found this on some class notes, which gives several different estimates of the error term, when going from the Riemann sum to its corresponding Riemann integral: $$\frac{b-a}{n}[f(b)-f(a)]$$ Does ...
1
vote
3answers
78 views

Show that this difference goes to zero,

$$\frac{1+\sqrt{2} + ... + \sqrt{N}}{N} - \frac{2}{3}\sqrt{N} \to 0.$$ The hint given in the question is this: choose appropriate Riemann sums and estimate the approximation error. My current work: ...
2
votes
0answers
30 views

Green's theorem application

Problem Determine all the circumferences $\mathcal C$ on $\mathbb R^2$ such that $$\int_{\mathcal C}-y^2dx+3xdy=6\pi$$ My attempt at a solution If I call $P(x,y)=-y^2$ and $Q(x,y)=3x$, then I can ...
1
vote
1answer
27 views

Integrate area of function over a tetrahedron

I actually attempted to enlist my professor help on this problem, but what he said was quick and I must not have written everything down because I cannot understand how this problem is supposed to be ...
0
votes
1answer
40 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(cos^{4}t+sin^{4}t)dt$

If $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, dt$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I tried Newton ...
0
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1answer
39 views

Evaluations of a Definite Integral with cosine function

How do you evaluate this integral? Does it involve an elliptical integral? What technique do I use to evaluate this integral? $$\int _{ 0 }^{ 2\pi }{ \sqrt { 5-4\cos { \theta } } d\theta } $$
3
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1answer
27 views

Convergence of Integrals of Exponential Functions

Let $f$ be a non-negative real valued function on $[a,b]$, and let $p:[a,b]\to(1,\infty)$ such that $f^p\in L^1([a,b])$. Let $p_n:[a,b]\to(1,\infty)$ be a (uniformly bounded) sequence of ...
0
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0answers
19 views

Rankine Hugoniot, taking limits

I have seen two different derivations of the Rankine Hugoniot jump conditions across a shock s(t) in the xt-plane. I present a summary of the two different derivations and then post my question in ...
0
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0answers
24 views

Yet another asymptotic series that needs to be analyticaly extended

Let $A>0$ and $1\le \mu \le 2$. Consider a following definite integral: \begin{equation} {\mathcal I}(A,\mu) := Re\left[\int\limits_0^\infty e^{-(k A)^\mu}\frac{\left(\gamma+\Gamma(0,\imath ...
0
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1answer
23 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
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0answers
24 views

To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) f_y$ exists.

Let $f : \Bbb R^2 → \Bbb R$ be defined by $f(x, y) := x^2 + y^2$ if $x$ and $y$ are both rational, and $f(x, y) := 0$ otherwise. To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) ...
1
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1answer
27 views

Doubling measure of an annulus

Recall that a doubling measure is a measure with the additional requirement that: $$\mu(B_{2R})\le C_\mu \mu(B_R)$$ for some contstant $C_\mu$. While solving some esercises related to doubling ...
6
votes
0answers
45 views

How to solve this definite Integral containing $E_{1}${.}!

The integral is: $$\int_{N}^{\infty}\frac{E_{1}(cz+d)}{az+b}e^{-pz}dz$$ where, $E_{1}${.} is the exponential integral, and $$a>0,\ b>0,\ c>0,\ d>0,\ p>0,\ N>0.$$ This is similar ...
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2answers
48 views

How to calculate an elementary integral

How do you calculate $$\int\dfrac{2 du}{(u^2+1)^2}$$ It does not seem too difficult but I do not know which method to use.
5
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1answer
81 views

Closed-form of $\int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

I've conjectured the following closed-form: $$ I = \int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx = -2\,G\,\ln2, $$ where $G$ is ...
1
vote
1answer
23 views

Find the region of integration as defined by two paraboloids

I've been given the following problem, and I'm completely unsure how to go about solving it. $$ \text{Find the volume of the solid enclosed by the}\\ \text{paraboloids } z = 16 \left( x^{2} + y^{2} ...
0
votes
1answer
25 views

How to evaluate dh/dt giving dV/dt?

Water evaporates from an open bowl of unspecified shape at a rate proportional to the area ofthe water surface; that is, $$\frac{dV}{dt} = -cA(h)$$ where V is the volume of water, A(h) is the area of ...
0
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2answers
65 views

Help with Calculate integral

Find $\int^a_0 \dfrac{3x^2-ax}{(x-2a)(x^2+a^2)} dx$ I tried using partial fractions and the substitution $u=a-x$ but I haven't made any real progress. Please help.
4
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1answer
64 views

Calculation of integral using two different methods? [on hold]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...
0
votes
2answers
23 views

Finding the bounds for a triple integration

I'm currently working on a problem stating: $\iiint_Q y*dV$, where Q is the solid that lies between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$, above the xy-plane, and below the plane z=x+2. My ...
3
votes
3answers
186 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
0
votes
1answer
17 views

Find area of exponential function over box-like region

The problem doesn't seem like it should be too difficult, I have a box-like region $B$ defined as: $$ \begin{align} 0 \le x \le 1\\ 0 \le y \le 3\\ 0 \le z \le 2 \end{align} $$ And the function to ...
-2
votes
2answers
35 views

How to solve this probability formulation? [on hold]

$\int_{200}^{250} P(a=x \land 450-x \leq b \leq 250)\space dx$, where $a$ and $b$ are uniformly distributed random variables on $(0,250]$ and $(10, 250]$ respectively.
0
votes
2answers
42 views

Integration by Partial Fractions $\int\frac{1}{(x+1)^3(x+2)}dx$

I'm trying to do a problem regarding partial fractions and I'm not sure if I have gone about this right as my answer here doesn't compare to the answer provided by wolfram alpha. Is it that I can't ...
2
votes
1answer
51 views

Finding the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$

I need to find the area under the cycloid $x=t-\sin (t),\;y=1-cos (t)$ above axis and between $x=0,x=2\pi$ using $\underline{\text{Green's theorem}}$ I found in Wikipedia this evaluation: ...
1
vote
1answer
47 views

Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
2
votes
2answers
46 views

Finding the volume of a solid bounded by a sphere and a paraboloid

I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 2$ and the paraboloid $x^2 + y^2 = z$. I know that to do this, I must use triple ...
6
votes
3answers
137 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
0
votes
1answer
61 views

Evaluate $\oint_{C}xy^2dx+2x^2 dy$

$$\oint_{C}xy^2dx+2x^2y dy$$ triangle:$$C=\{(0,0),(2,2),(2,4)\}$$ My attempt: Using Green's theorem $$\oint_{C}\underbrace{xy^2}_{P}dx+\underbrace{2x^2y}_{Q} dy=\iint\bigg(\frac{\partial ...
6
votes
1answer
41 views

Limit behavior of a definite integral that depends on a parameter.

Let $A>0$ and $1\le \mu \le 2$. Consider a following integral. \begin{equation} {\mathcal I}(A,\mu) := \int\limits_0^\infty e^{-(k A)^\mu} \cdot \frac{\cos(k)-1}{k} dk \end{equation} By ...
1
vote
2answers
31 views

How to find the surface area of revolution of an ellipsoid from ellipse rotating about y-axis

Suppose the ellipse has equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. I understand the way to obtain the surface area of the ellipsoid is to rotate the curve around y-axis and use surface of ...
0
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1answer
28 views

Change of variables when integrating over a triangle

I want to calculate the integral $$ \iint_D(x-y)dxdy $$ where D is the triangle made up of the vertices (0,0), (-2,1) and (-1,3). (Graph) My idea was to do this substitution $$ \begin{equation} ...
19
votes
3answers
155 views

Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$$ I simplified the limit to $$\dfrac{1}{2}\lim_{n \to \infty} \int_{0}^{\frac{1}{2}} ...
1
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0answers
20 views

Error estimate of polynomial quadratures missing some terms

Normally, for trapezoid rule and simpson's rule, etc, error analysis is done by using the error formula for interpolation. However, if the polynomial is restricted to some terms, for example, a ...
-1
votes
0answers
32 views

Quick Integration by parts question [on hold]

If while doing integration by parts I get a sum of +infinity and -infinity, can I obtain that the Integral diverges?
0
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2answers
63 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
5
votes
2answers
105 views

The value of the integral $\int_0^2\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$

The value of definite integral $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$ is $$(A)\,4 \quad(B)\,5 \quad (C)\,6 \quad(D)\,7$$ My attempt: I tried using ...
0
votes
1answer
58 views

Evaluate $\iint dydx$ on the domain $0\leq r\leq1$, ${\pi}/{3}\leq\theta \leq{2\pi}/{3}$ [duplicate]

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}\;\;\;\;,0\leq r\leq1$ $\color{blue}{\text{without using polar ...
0
votes
2answers
60 views

Certain integration technique

What technique to follow when integration functions in the form: $$\sin ax\over \sin bx$$ $$\cos ax\over \cos bx$$ $$\sin ax \over \cos bx$$ I do believe that all these forms should have a similar ...
0
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1answer
31 views

Writing line integral as 1-form

If $F: \Bbb R^n \rightarrow \Bbb R^n $ is a vector field and $\phi : [a,b] \rightarrow \Bbb R^n$ is a continously differentiable path we defined the integral of $F$ along $\phi$ as $\int_{\phi} F = ...
2
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1answer
40 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
0
votes
2answers
42 views

Probability and integration

Compute $E[e^{tX}]$ where $X ∼ \mathcal{N} (0, 1)$. [Hint: Complete the square in the exponent.] Do we set up the integral from $0$ to $1$? Then how do you solve this integral?
14
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3answers
392 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
2
votes
3answers
81 views

$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$

Prove that for all $\xi \in \mathbb{C}$, $$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$$ I don't really know how to compute this integral. Can you please help me?
0
votes
1answer
83 views

Finding the area under the curve $y=3-3\cos(t),x=3t-3\sin(t)$

I need to find the area under the curve $\color{blue}{y=3-3\cos(t),x=3t-3\sin(t)}$ and between $\color{blue}{x=2\pi,x=0\text{, above axis}}$ using $\color{blue}{\text{Green's theorem}}$. My attempt ...
6
votes
2answers
124 views

Integrating $\frac{x^3}{\exp(x)-1}$ from $0$ to $\infty$

While doing Physics and trying to prove the law of Stefan-Boltzmann from Plancks-law one comes to the integral \[ \int_0^\infty \frac{x^3}{\exp(x)-1} \mathrm{d}x=\frac{\pi^4}{15} \] and I would like ...
1
vote
2answers
25 views

Finding the flux of $\iint \vec F\hat n\;ds$

I need to find the flux $\displaystyle\iint \vec F\hat n\;ds$ of the vector feild $\vec F=4x \hat i-2y^2\hat j+z^2 \hat k$ throughe the surface $S=\{(x,y,z):x^2+y^2=4,z=0,z=3\}$ My attempt: (I'm ...
2
votes
1answer
40 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
2
votes
0answers
35 views

Help solving integration: $I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(a/\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}\right)dx$

My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$ $$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$ I ...
1
vote
3answers
85 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...