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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-4
votes
1answer
14 views

Triple Integrals

Find the volume of the region bounded by $(x^2+y^2+z^2)^2=x.$ I'm having issues setting my bounds (specifically $\theta$) so far I have $0<r< [\sin(\phi) \cos(\theta)] ^{(\frac{1}{3})}$ ...
0
votes
0answers
14 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
2
votes
2answers
25 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
7
votes
5answers
141 views

Evaluate $\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$

I want to find a quick way of evaluating $$\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$$ This problem appeared on the qualifying round of MIT's 2014 Integration Bee, which leads me to think ...
0
votes
1answer
37 views

Volume of solid of rotation about x-axis

Rotation of the region bounded by $x=2y^2-1,\; x=y^2$ and $x$-axis about $x$-axis. I draw out the graph, and found intersection is at $(1,1)$ and $(-1,-1)$ So is it correct to continue doing by ...
1
vote
0answers
21 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
0
votes
2answers
20 views

About minimal curvature of splines

I am given a the following problem set: Let $s$ be a natural cubic spline that interpolates a function $f \in \mathcal{C}^2 ([a,b])$ at points $a = x_0 < x_1 < \ldots < x_n =b$ with ...
4
votes
1answer
29 views

Convergence of multiple integral in $\mathbb R^4$

Denote $(x,y,z,w)$ the euclidean coordinates in $\mathbb R^4$. I am trying to study the convergence of the integral $$\int \frac{1}{(x^2+y^2)^a}\frac{1}{(x^2+y^2+z^2+w^2)^b} dx\,dy\, dz\, dw$$ over a ...
0
votes
1answer
46 views

Finding volume of convex polyhedron given vertices

I am trying to compute the volume of the convex polyhedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,2,0)$, $(0,0,3)$, and $(10,10,10)$. I am supposed to use a triple integral but am struggling with how ...
0
votes
0answers
10 views

How to partially differentiate an integral with a density function?

I am given this result: $$\frac{\partial}{\partial x(t)} \lambda \int u(x(t)) f(t) \mathrm{d}t = \lambda u^\prime(x(t)) f(t)$$ How do I arrive to this result? Where am I mistaken? Using the chain ...
2
votes
0answers
21 views

Implementing the Risch algorithm to integrate $\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}$

Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we ...
0
votes
0answers
15 views

Find the volume of revolution about x-axis

I need to find the volume of the solid generated by revolving a circle about the x-axis. The equation of the circle is given by $$x^2 + (y - 1)^2 = 1$$ indicating that it is centred at $(0, 1)$ with ...
1
vote
1answer
137 views

Calculating the variance, mean, and autocorrelation of a time series.

How can I calculate the mean, variance, and autocorrelation function: $$Y_t=5+Z_t+ 0.6Z_t-1$$
3
votes
0answers
104 views

Is there a generalization of integration by parts?

In the original integration by part formula there are two functions $u(x)$ and $v(x)$. What if the integral involves another function $w(x)$ as well? Second of all, I know that there is a several ...
-1
votes
2answers
31 views

Is this integral correctly calculated?

The problem is that I can't use wolframalpha to check this because he is worried about integration limits: I have $a>0$ and $t \in (-1,1).$ $$(1-t)^{\frac{a}{2}} \int_0^t \frac{1}{(1-x)^{a+1} }dx= ...
0
votes
4answers
57 views

Arc-length of a Logarithmic Spiral

I want to calculate the arc-length of the Log spiral given by the equation: $\vec{x}(t)=\begin{pmatrix} e^{-\alpha t} \cos t \\ e^{-\alpha t} \sin t\end{pmatrix}$ $\alpha >0$ and $t \in [0,T]$ ...
0
votes
1answer
27 views

How to show this integral (Error function)

I'm given this question. Show that $$\int_{0}^{0.25}\frac{1}{\sqrt{x}}e^{-x}dx=\int_{0}^{0.5}2e^{-u^2}du$$. As I know this integral is an error function. How to show? Can anyone give me some hints? ...
0
votes
1answer
1k views

Find an expression for the area under the graph of f(x) as a limit?

$f(x) = \frac{2x}{x^2 +1}, 1 \leq x \leq 3$ Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the ...
1
vote
1answer
21 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
4
votes
1answer
64 views

Why are the differentiation/integration rules what they are?

So I understand what rules you use where, and the general forms of the rules like: $$\left(\frac{d}{dx}\right)^nx^k=\frac{k!}{(k-n)!}x^{k-n}$$ My question is why are these the formulas that give us ...
1
vote
1answer
399 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
0
votes
2answers
76 views

Evaluate $\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$

My question is: Evaluate $$\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$$ for $k=0,1,...,(n-1)$ and $n \in \mathbb{N}$. I've tried integration by parts but without much success. Any ...
2
votes
3answers
422 views

integrating using student t distribution

Evaluate the integral $\int_0^\infty\frac{1}{1+x^2}dx$ using the Student t distribution. I don't know where to start. I am assuming that I can't just do regular integration. I don't know how I am ...
1
vote
1answer
31 views

Double Integral Change of variable help

I am having some trouble getting this problem set up, and would appreciate any help. Problem: $\iint \frac{1}{(x+y)^2} dA$. Integrate using change of variables over the region inside the lines ...
1
vote
1answer
24 views

Double integral variable change help

I'm having a tough go with this problem. $\iint \frac{x^2}{y^3} dA$ , Integrate using a change of variables over the region defined inside the curves $y=2x,\; y=x,\; y=x^2,\; y=2x^2$ . I graphed it ...
0
votes
1answer
53 views

Primitive of the function $(\sin x)/x$

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive. Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?
0
votes
0answers
22 views

Renewal equation

I have a question on renewal equation. $$m(t) = F(t) + \int_0^t (m(t − x)dF(x))$$ can someone tell me how to compute the integral part? it's better if you can tell me step by step until we get the ...
0
votes
0answers
33 views

Hardy Littlewood Circle Method

I'm working through Vaughan's book on the Hardy Littlewood circle method, which uses the following lemma: Suppose that $\alpha \geq \beta$ are positive real numbers, and that $\beta \leq 1$. Then: $ ...
0
votes
1answer
1k views

proof of the second generalized mean value theorem for integrals

Let $f,g,g´$ be continous on $[a,b]$ and $g$ monotone on $[a,b]$; then there exist $c\in (a,b)$ so that $$\int_{a}^{b}f(x)g(x)dx=g(a)\int_{a}^{c}f(x)dx+g(b)\int_{c}^{b}f(x)dx$$ Ineed to apply the ...
5
votes
5answers
257 views

How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$

This problem appears at the end of Trig substitution section of Calculus by Larson. I tried using trig substitution but it was a bootless attempt $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
8
votes
2answers
110 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
0
votes
4answers
62 views

Evaluating $\int_{-a}^{a}\sqrt{a^2-x^2}dx$

Question: How to evaluate $$\int_{-a}^{a} \sqrt{a^2 - x^2} dx$$ This came up while trying to prove that the area of an ellipse is give by $\pi a b$ where $a$ and $b $are the major and ...
4
votes
4answers
84 views

How to evaluate $\lim _{n\to \infty }\:\int _{1/(n+1)}^{1/n}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $$\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx,\:n\in \mathbb{N}$$ First step I wrote that $\int ...
0
votes
1answer
38 views

The existence of anti-derivatives

The only thing I can think of is that the function is continuous hence the anti derivative exists. I was wondering if there is anything else that needs to be done/said?
1
vote
1answer
36 views

Convergence of $\int_0^1 \frac{\ln(1-x)\sqrt{x-x^2}}{\sin(\pi x)} \, dx$

I have difficulties with convergence of this integral: $$\int_0^1 \frac{\ln(1-x) \sqrt{x-x^2}}{\sin(\pi x)} \, dx$$ I found similar problem here Covergence of integral but I don't get the solution ...
2
votes
1answer
23 views

Green's Theorem and limits on y for flux

I'm working through understanding the example provided in the book for the divergence integral. The theorem (Green's): $$ \oint_C = \mathbf{F}\cdot \mathbf{T}ds = ...
2
votes
2answers
793 views

Prove using integration that $polygon → circle\space \text{as}\space number\space of\space sides → infinity$

Say we have a regular polygon $s$, with number of sides $n$: Is there a way to prove that as $n → ∞,\space $then $s → circle$ using integration?
0
votes
3answers
57 views

Problem in indefinite integral. (Exponential)

I'm given this integral to integrate. I've no idea where to start with. Perhaps someone can give me some hints or guide me. Thanks a lot. $$\int\frac{(x^3)e^{x^2}{}}{x^2+1}dx$$
7
votes
0answers
55 views

Integral Inequality $\leq n^{3/2}\pi$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $ \int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$ It's easy to see ...
0
votes
0answers
11 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
1
vote
0answers
19 views

How would i find the volume of a cone in the $interval [0,a]\times[0,a]\times[0,a]$ and how it's surface area? (using integration?)

whichEssentially i want to find the measure of $z^2\leq x^2+y^2$ and $z^2=x^2+y^2$. Now i know for one of them i would incorporate cilindrical coordinates: $$g(r,\phi,z)=(rcos \phi, r sin\phi,z)$$ ...
3
votes
1answer
35 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't place nicely with ...
0
votes
2answers
20 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
0
votes
3answers
41 views

Struggling with integration/differentiation

quick question as I'm sure this is simple but it has me stumped. I have to integrate and differentiate this equation. Not sure on the exponential, had a couple of goes but it doesn't look right. ...
1
vote
0answers
41 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
-3
votes
2answers
30 views

Proving that a particular integral is a rational number [on hold]

I have the sequence $\left\{ I_n \right \}_{n\geq 0}$ given by: $$I_n=\int_{0}^{1}\frac{\left ( x^2+x+1 \right )^n - 1}{x^2+1}\,dx$$ and I have to prove that $I_{4n+1} \in \mathbb{Q}$ .
1
vote
1answer
37 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
0
votes
1answer
28 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
0
votes
1answer
12 views

Reasoning behind method of steepest descent

I am considering the method of steepest descent from my notes. I have written that $$\int_a^b dx e^{g(x)} \sim e^{g(x_0)} \int_{\infty}^{\infty}dx \exp \left[-\frac{1}{2}(x-x_0)^2|g^"(x_0)|\right] ...
1
vote
1answer
49 views

Find this integral $I(x)=\int_{0}^{+\infty}\frac{1}{y}e^{-y-\frac{x}{y}}dy$

Find this integral $$I(x)=\int_{0}^{+\infty}\dfrac{1}{y}e^{-y-\dfrac{x}{y}}dy$$ I think $$I'(x)=-\int_{0}^{+\infty}\dfrac{e^{-y-\frac{x}{y}}}{y^2}dy$$ Now I have no idea of how to continue