All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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When $C$ is the positively oriented circle centered at 0 with a radius of 2, what is $\int_C \dfrac{1}{z^3+4z^2+3z}dz$?

I am reviewing for a complex analysis final and this was a question on the review sheet. No answers were provided so I attempted it on my own. Using Cauchy's Integral Formula, I have that $$2\pi i ...
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1answer
46 views

Showing integral is bounded

Let $A \subset \mathbb{R}^n$ be open and bounded. Is it true that $$\int_A \int_A |x-y| \, dx \, dy \leq R|A|^2$$ where $R$ is some number (eg. the radius of a ball containing $A$) and $|A|$ is ...
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0answers
42 views

Integrability and product measure

Let $X$ and $Y$ be subsets of $\mathbb{R}$, and let $\mu$ be a measure on $X$ and $\nu$ a measure on $Y$. Let $f : X \times Y \rightarrow \mathbb{R}$ be $\mu$-summable and $\nu$-summable, i.e. ...
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0answers
29 views

Evaluate spatial variation of density-like scalar

Apologies if this has been asked previously, but I'm not totally sure of the best way to pose the question. Background I'm evaluating the variation of a spatially varying scalar field $p$ ...
2
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1answer
57 views

evaluation of integral $\int_{-\infty}^{\infty}e^{-b\sqrt{x^{2}+a}}dx$

I need to find the analytical expression of the following integral when $a,b > 0$: $$\int_{-\infty}^{\infty}e^{-b\sqrt{x^{2}+a}}dx.$$ Can someone help me here?
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4answers
104 views

Inequality $\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$

Let $f$ be a positive continuous function defined on a closed interval $[0,1]$, then it is true that: $$\left(\int_0^1 f(x)dx\right) \left(\int_0^1 \frac{1}{f(y)} dy\right) \ge 1$$ I tried to show ...
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1answer
52 views

Integration of a function which behavior is comparable with the function $\sin$.

I am struggling with the following problem: Let $$\frac{\pi_p}{2}=\int_0^1\frac{1}{(1-s^p)^{1/p}}ds$$ where $p\in (1,\infty)$. Define $\operatorname{sin_p}:[0,\pi_p/2]\to [0,1]$ by ...
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1answer
54 views

how to represent this domain of integration

I have an exercice from a Stewart's book, I don't have the book with me and I don't remember the number and the page... so the question is to evaluate : $$\int_{1/ \sqrt 2}^1 \int_{\sqrt{1-x^2}}^x xy ...
3
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1answer
76 views

Integral involving a confluent hypergeometric function

I have the following integral involving a confluent hypergeometric function: $$\int_{0}^{\infty}x^3e^{-ax^2}{}_1F_1(1+n,1,bx^2)dx$$ where $a>b>0$ are real constants, and $n\geq 0$ is an ...
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1answer
45 views

Can you help integrate this Gaussian?

Mathematica is playing up and I need: $$ \int_{0}^{a}\exp\left(-\rho\left(\frac{\sqrt{2}}{b\sqrt{r}}x^{1/2}-\pi ...
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1answer
33 views

Variation on exponential pdf expected value: integration

I know that $$ \int_{0}^{\infty}t \mu {\rm e}^{-\mu t}\,{\rm d}t = {1 \over \mu^{2}}\,\qquad \mu > 0 $$ I haven't been able to figure this out with substitution: $$ \int_{0}^{\infty}t^2 \mu {\rm ...
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0answers
40 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
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votes
3answers
121 views

Find value of integral: $I=\int_0^{2\pi}\frac{dx}{(2+\cos x)^2}$

Find value of integral: $$I_1=\int_0^{2\pi}\frac{dx}{(2+\cos x)^2}$$ and $$I_2=\int_0^{2\pi}\frac{dx}{(2+\sin x)^2}$$ I don't know how, i need a solution, please
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2answers
3k views

Finding volume of a frustum of a pyramid

I need to find the volume of a frustum of a pyramid with square base of side $b$, square top of side $a$, and height $h$(using integrals). I have no idea how to do questions like these, I only know to ...
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1answer
75 views

Determine value of integral:$I=\int_0^1\frac{\ln(1+x)}{x}dx$

Determine value of integral:$$I=\int_0^1\frac{\ln(1+x)}{x}dx$$ I use Taylor's expansion with $x_0=0$, we have: $$\ln(1+x)=\sum_{i=1}^{\infty}\frac{(-1)^{i+1}x^i}{i}$$ Hence ...
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4answers
113 views

Why is integral $\int \cos (t-u)\, du = -\sin(t-u)$

$$\int \cos(u)\,du= -\sin(u) + C $$ But why is, $$\int \cos (t-u)\,du=\ -\sin(t-u)$$ and not $\sin(t-u)$?
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1answer
62 views

Conservative Vector Fields

One of the theorems for a vector field to be conservative is that $$\frac{\partial N}{\partial x}=\frac{\partial M}{\partial y}$$ for $$F=\langle M,N\rangle.$$ To find the $$\int ...
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1answer
33 views

Integral over two sets when the measure of their symetric difference equals zero

Let $A$ and $B$ be two measurable sets of $X$. Show that if $m(S(A,B))=0 $ then $$\int_{A} f\, dm =\int_{B} f\,dm,$$ where $S(A, B)$ is the symetric difference. my method was to prove that ...
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2answers
86 views

How find this $I=\iint_{\Sigma}(x^2+y^2+z^2)^{-\frac{3}{2}}(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4})^{-\frac{1}{2}}dS$

Find this Surface integral $$I=\iint_{\Sigma}(x^2+y^2+z^2)^{-\frac{3}{2}}\left(\dfrac{x^2}{a^4}+\dfrac{y^2}{b^4}+\dfrac{z^2}{c^4}\right)^{-\frac{1}{2}}dS$$ where ...
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1answer
45 views

Transforming a series to an integral with respect to counting measure

I'd really appreciate it if somebody could help me understand why we have this with a step-by-step explanation (i.e. in an argument complete way) : $$ \sum_{k=1}^{n} {\frac {n} {k^2+nk+1}} = \int ...
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1answer
44 views

Passing limit inside integral for functions in $L^1+L^2$ norm

Let $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, and let $f_k$ be functions in the Schwartz class such that $\|f-f_k\|_1+\|f-f_k\|_2\rightarrow 0$ as $k\rightarrow\infty$. Define ...
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1answer
74 views

Integrating With Trig Subsitution

Can someone please explain how to calculate the integral of $\frac{\sqrt{1 + x^2}}{x}$ using trig substitution?
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1answer
244 views

Use Fourier transform to calculate double integral of harmonic function

Let $$P_y(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty e^{-y|t|}e^{ixt}dt=\dfrac{1}{\pi}\dfrac{y}{x^2+y^2}.$$ Then $P_y(x)$ is harmonic in the upper half-plane $y>0$ and for $f\in L^1(\mathbb{R})$, ...
0
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1answer
14 views

Why is $\int_{A}{\frac{y^2}{r_0-y}dA}=\int_{A}{-y dA}+r_0\int_{A}{\frac{y}{r_0-y}dA}$?

My textbook on Advanced Mechanics of Solids makes the following substition $$\int_{A}{\frac{y^2}{r_0-y}dA}=\int_{A}{-y dA}+r_0\int_{A}{\frac{y}{r_0-y}dA}$$ The context is curved beams, but I don't ...
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3answers
154 views

Calculate $\int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)} \operatorname d\!x$

Calculate$\displaystyle\int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)}dx$, where $a,b\in\mathbb{R}$ $\bf{My\; Try}:$ Let $\displaystyle I=\int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)}dx$ Now Let ...
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1answer
22 views

Properties of integration with a degree 2 polynomial

In my textbook, there is a true/false question: $ \int_{-5}^{5}(ax^2+bx+c)dx = 2\int_{0}^{5}(ax^2+c)dx $ Solving this particular case I found both sides to equal: $ \frac{250a}{3} +10c $ I ...
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3answers
62 views

Properties of integrals

How do I prove that the integral of a product is not equal to the product of integrals? $ \int_{a}^{b} f(x)g(x)dx \neq \left(\int_{a}^{b} f(x)dx\right)\left(\int_{a}^{b}g(x)dx\right) $
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0answers
24 views

Conservativeness on a graph

I'm trying to build a conservative vector field out of something smaller than $\mathbb{R}^2$ to understand how the "conservative" property of differences-of-scalar-fields leads to Green's theorem. (In ...
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1answer
40 views

Use spherical coordinates to evaluate

Use spherical coordinates to evaluate $\int_{-2} ^{2} \int_0 ^{\sqrt{4-y^2}} \int_{-\sqrt{4-x^2-y^2}} ^{\sqrt{4-x^2-y^2}} y^2\sqrt{x^2+y^2+z^2} dz \ dx \ dy$ I did like this. Is that right ? ...
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1answer
58 views

Find multiple integrals $I_{\max}(k,n)$ and $I_{\min}(k,n)$ in various ways

$I_{\max}(k,n)=\underbrace{\int\limits_0^1\int\limits_0^1\dots\int\limits_0^1}_k\left(\max\limits_{1\le i\le k}x_i\right)^n\,dx_1dx_2\dots dx_k$ ...
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2answers
31 views

Doing a Line Integral Problem

Here is my attempt: $$W=\int_C\vec{F}\cdot d\vec{r}\\=\int_C\frac{\alpha x}{(x^2+y^2)^{3/2}}dx+\frac{\alpha y}{(x^2+y^2)^{3/2}}dy\\Using\quad x=2t+1\quad and \quad y=-2t\quad for\quad 0\le t\le ...
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3answers
182 views

Changing from Cartesian coordinates to Polar coordinates

Rewrite the iterated integral $$\int_0^1 \int_0^{\sqrt{2y - y^2}} (1 - x^2 - y^2)\,dx\,dy$$ in polar coordinate form. Do not evaluate the integral. Here is my answer: ...
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3answers
121 views

Calculating the area between graphs of $x^2$ and $x + 1$

This is actually a question from Apostol's Calculus book (find it on p. 94). I would like to know if my reasoning is reasonable. I need to calculate the area between graphs of $f(x) = x^2$ and $g(x) ...
3
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2answers
72 views

Integrals: Find the area of the region.

Find the area of the region in the first quadrant bounded by $y=x^2$, $y=4x^2$, and $y=2$. The answer given is $\frac{2}{3} \sqrt2$ but I have no idea what this question is asking. I'm used to ...
2
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1answer
62 views

Problems with integrals, polar coordinates.

I am having problems parameterizing these integrals: $$\int_A{\frac{x}{1+x^2+y^2}}\mathrm{d}x\mathrm{d}y$$ for $A = \mathbb R^2 \bigcap \,\{y \ge 0\}$ and the volume of $M = \{(x, y, z) \in \mathbb ...
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4answers
106 views

How to integrate this double integral?

$$\iint \limits_D 2x^2e^{x^2+y^2}-2y^2e^{x^2+y^2} dydx $$ where D is the region $x^2+y^2=4$ I tried changing it to polar, but it didn't make any use. $\iint \limits_{D(r,\theta)}2r^3\cos2\theta ...
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0answers
29 views

Interchanging $\Re$ and integral sign

if a function $f$ is differentiable, and $\int_{-\infty}^{\infty}|f(t)|^2\,dt<\infty$. Is it true that ...
3
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1answer
55 views

$\lim_{n \to \infty}(1/(n^3+1)+1/(n^3+2)+…+1/5n^3)=\ln5$

How do I show that $$\lim_{n \to \infty}(1/(n^3+1)+1/(n^3+2)+...+1/5n^3)=\ln 5$$ I know this can be done using an integral but for this particular question I cannot simply find an equivalent Riemann ...
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1answer
44 views

why does using the inverse give the same answer of this integral.

Find the area bounded by $x=4-y^2$ and $x=y-2$ $A = \int_{-5}^0 \left[\left(x+2\right) + \sqrt{4-x}\right]\,\mathrm{d}x + \int_0^4 2\sqrt{4-x}\,\mathrm{d}x$ This is the solution in the book. ...
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0answers
50 views

Is sgn(x) Riemann integrable?

sgn(x) has a jump discontinuity at zero but is continuous everywhere else. So is sgn(x) Riemann interable on any given [a,b]?
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2answers
381 views

If f is of bounded variation is f Riemann integrable?

I want to know if f is of bounded variation on [a,b] does it follow that f is Riemann integrable on [a,b]?
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1answer
36 views

Existence of function that blows up on a null set

Let $m$ denote Lebesgue measure in $\mathbb{R}^d$. Show that for any set $E$ with $m(E)=0$, there is a non-negative integrable function $f$ such that $$ \liminf_{m(B)\to 0}\frac{1}{m(B)}\int_B f(y)\ ...
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1answer
42 views

Order of Integration E[X]

I understand that $\int_0^\infty P(X>x)dx=E[x]$, and also the logic behind the discrete version here. What I don't understand is how the limits of integration change as is seen here, from $(x, ...
1
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2answers
99 views

How can these two be equivalent (wolfram-alpha incorrect) !?

So wolfram-alpha reads The integral of $$\int \frac{1}{\sqrt{a^2-x^2}}dx=\tan^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right)$$ but that $$\int\frac{1}{\sqrt{a^2-x^2}}dx \;\mathrm{where}\; a=5 ...
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0answers
31 views

Intuition behind precision with quadratures

From the theorems in my book, I know that: If for some interger $m$, $Q_n(f)=I(f)$ for all $f(x)\in \mathscr{P_n}$ then we say that the quadrature has precision $m$. Also that if a quadrature $Q_n$ ...
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votes
1answer
90 views

Find $f(x)$ for the values of $x$ and $f'(x)$ given

Given the values of the derivative $f′(x)$ in the table and that $f(0)=150$, find or estimate $f(x)$ for $x=0,2,4,6$. $x: 0, 2, 4, 6$ $f'(x): 10, 24, 37, 47$
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3answers
1k views

Find the area bounded by the curves $y=e^x$, $y=xe^x$, and $x=0$

Find the area bounded by the curves $y=e^x$, $y=xe^x$, and $x=0$ I know how to solve the integral for this, but I'm getting hung up trying to find the points of intersection for the two equations. ...
2
votes
5answers
210 views

Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$

Suppose $f$ is a continuous single-variable function, prove that: $$2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$$ This question was just on my Calculus III final ...
2
votes
1answer
90 views

Do The Integrals Tend to 0?

Consider the integrals $\int_1^\infty \frac{k}{x^2+k^p\cos^2x}dm(x),$ where $m$ is the Lebesgue measure. For what $p$ do the integrands have an integrable majorant? For what $p$ do the integrals tend ...
0
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0answers
42 views

Bounding a convolution with a maximal function

Consider a family of kernels $\{K_{\epsilon}\}_{\epsilon>0}$ such that: $\int_{\mathbb{R}^d}K_{\epsilon}\ dx=1$ $|K_{\epsilon}(x)|\leq A\delta^{-d}$ for all $\delta>0$ $|K_{\epsilon}(x)|\leq ...