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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2
votes
2answers
32 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...
8
votes
1answer
91 views

Integration validity of $\int\frac{1}{\sqrt{a^2 + x^2}}\,dx$

I'm just wondering if the following integration is valid. \begin{array}{l} \int {\frac{1}{{\sqrt {{a^2} + {x^2}} }}} dx\\ {\rm{Let }}{u^2} = {a^2} + {x^2}\\ 2udu = 2xdx\\ \frac{{du}}{x} = ...
0
votes
0answers
20 views

How do you find the volume of a function rotated about the x axis along it's derivative? [on hold]

So when you rotate a function, it is usually vertically. How do you rotate it around it's derivative, assuming that volume/area can overlap?
4
votes
4answers
93 views

Am I getting the right answer for the integral $I_n= \int_0^1 \frac{x^n}{\sqrt {x^3+1}}\, dx$?

Let $I_n= \int_0^1 \dfrac{x^n}{\sqrt {x^3+1}}\, dx$. Show that $(2n-1)I_n+2(n-2)I_{n-3}=2 \sqrt 2$ for all $n \ge 3$. Then compute $I_8$. I get an answer for $I_8={{2 \sqrt 2} \over 135}(25-16 ...
-4
votes
2answers
29 views

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $H(x) := \int_x^b f \ \ ,x\in I.$ [on hold]

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $$H(x) := \int_x^b f, \ \ x\in I.$$ To find $H'(x)$ for $x \in I.$ I am stuck with the problem please help.
1
vote
0answers
34 views

Integral of a function which is everywhere discontinuous?

Yesterday, I tried to carry out a little thought experiment when it came to taking limits and have found that it has pushed my understanding of them to the breaking point. I tried considering the ...
3
votes
2answers
76 views

Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
-2
votes
0answers
23 views

Integral Proof: Integral between h and 0, (h - z)(z - l)dz [on hold]

How does the following give the result of l = h/3: Integral between h and 0, (h - z)(z - l)dz
1
vote
1answer
43 views

Double Integration word problem

In a certain metropolitan area, the population is approximated by the function: $$P(x,t)=\frac{\ 7274e^{0.5t}}{1+x}$$ Where $x$ is the number of miles from the center of the city, and $t$ is the ...
1
vote
0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
1
vote
1answer
42 views

Evaluate Double Integration

Evaluate $\iint−3x^2 dA$ over the region in the first quadrant bounded by the hyperbola xy=16 and the lines $y=x$, $y=0$, and $x=8$. I have drawn a picture, but I am still a little unsure on what to ...
2
votes
1answer
112 views

How can I prove the integral $ \int_{1}^{x} \frac{1}{t} \, dt $ is $\ln x $ with this approach?

I have been trying to find a proof for the integral of $ \int_1^x \dfrac{1}{t} \,dt $ being equal to $ \ln \left|x \right| $ from an approach similar to that of the squeeze theorem. Is it possible to ...
1
vote
2answers
29 views

function such that the sum of previous f(x) is smaller than f(x)

Just out of curiosity: is there a function $f$, such that $ \forall x, \sum_{x'<x} f(x') < f(x) $ sum or integral...
-1
votes
0answers
28 views

Calculus 2 - Rotating a region about an axis

I am having issues with the disk method and shell method when rotating a region around an axis. Example: The region bounded by $y=x^{\frac{1}{3}}, x=4y$, axis $x=3$ I am thinking that shell method ...
0
votes
2answers
38 views

Product Integral: Integrability

Given measure spaces $X$ and $Y$. Then it holds: $$\int_Y\int_X|\eta(x,y)|\mathrm{d}\mu(x)\mathrm{d}\nu(y)<\infty\implies\int_X|\eta(x,y)|\mathrm{d}\mu(x)<\infty\quad(y\in Y)$$ Can this ...
-3
votes
1answer
41 views

how to solve this problem by easy way [on hold]

Can the following double integral: $$\iint xy(x+y+25)^{3/2} dx dy$$ be solved in an easy way?
-1
votes
2answers
37 views

Integration problem with $e$ and $\ln$

Can anyone help me solve this? I tried with integration with parts, but without luck. The function to be integrated is $$\frac{e^{x+\ln x}}{x}$$
0
votes
3answers
104 views

A simple looking integration : $\left(\frac{x^3}{1+x^5}\right)$

One of my friends gave me this problem about a week back and since then, I have been toiling to get a solution to this problem, but I just get stuck at some step. Can someone please tell me the steps ...
2
votes
5answers
98 views

Mistake in evaluating $\int\dfrac{dx}{\ln(x)}$

Evaluate: $$I=\int\dfrac{dx}{\ln(x)}$$ My attempt: $$$$ $$I=\int \dfrac{x'}{\ln(x)} dx$$Integrating by Parts,$$\dfrac{x}{\ln(x)}-\int\dfrac{x}{(\ln(x))'}dx$$$$=\dfrac{x}{\ln(x)}-\int ...
1
vote
0answers
32 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
-3
votes
2answers
37 views

Crazy and difficult Limits and integration

This limit take from me much time to solve and finally I can't. So please help me to solve.. Find $L$ $ L =\displaystyle\lim_{x\to \infty} \frac{\displaystyle\int_{1}^{x} t^{t-1} ( t + tln (t) +1 ) ...
0
votes
1answer
70 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
2
votes
1answer
31 views

Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface ...
0
votes
0answers
21 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
6
votes
1answer
43 views

$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I$ [duplicate]

Let $I:=[a,b]$ and let $g: I \to \Bbb R$ be continuous on $I$. Suppose that there exists $K > 0$ such that $$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I.$$ Then $g(x) = 0\ \ \forall x \in I $. ...
2
votes
5answers
76 views

if we have $(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$ then $f(x) =x \ \forall x\geq0$.

Let $f: [0, \infty) \to \Bbb R$ be continuous and $f(x) \neq 0 \forall x>0$. If we have $$(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$$ then $f(x) =x \ \forall x\geq0$. We have $(f(x))^2 = 2 ...
0
votes
1answer
11 views

$F(x) := (n- 1)x-\frac{ (n- 1)n}{2}$ for $x \in [n- 1, n), n \in \Bbb N$ using this result to evaluate $\int_a^b[x]dx.$

Let $F(x)$ be defined for $x \geq 0$ by $F(x) := (n- 1)x- (n- 1)n/2$ for $x \in [n- 1, n), n \in \Bbb N$. Show that $F$ is continuous and evaluate $F'(x)$ at points where this derivative exists and ...
6
votes
0answers
24 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
1
vote
0answers
21 views

On utilizing the Leibniz rule of integration on a non compact interval.

I am following some slides that you can find here. At slide $\approx$ 24 a problem arises, to find $$\DeclareMathOperator*{\argmin}{\arg\!\min} \argmin_{\hat{y} } -\int_{-\infty}^{\hat{y}} (y ...
3
votes
3answers
50 views

How to find: $\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$?

How to find: $$\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$$ I tried multiplying it all out but I just ended up in a real mess and I'm wondering if there is something I'm ...
2
votes
1answer
58 views

Show that there exist continuous functions $g,h:[0,1]\rightarrow \mathbb{R}$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a Riemann Integrable function. Let $\epsilon>0$. Show that there exist continuous functions $g,h:[0,1]\rightarrow \mathbb{R}$ such that $g(x)\leq f(x)\leq ...
3
votes
2answers
77 views

Evaluate the integral $\int \frac{x}{a+bx^3}\ dx$

How do I solve integral at this form $\displaystyle\int \frac{x}{a+bx^3}\ dx$ ? I have tried a lot of things, but it doesn't work. I also know that the solution isn't that easy.
1
vote
3answers
77 views

Integral Convergence $\sin{x}/x^{3/2}$

Does the following integral converge: $$\int_0^\infty{\frac{\sin x}{x^{3/2}}}dx$$ I have tried to integrate this by parts and arrived at: $$-x^{-3/2}\cos x -\int \frac 12{x^{-1/2}}\cos{x} dx $$ ...
0
votes
0answers
72 views

$\int_0^b \ln(\sin(ax))dx$ [duplicate]

Problem: Evaluate $$\int_0^b \ln(\sin(ax))dx$$ Unfortunately I have no idea as to how to proceed with finding a closed form for the above Integral. The $a$ in the integrand made me think of ...
1
vote
1answer
59 views

Find the area using double integral and polar coordinates.

I need to find the area using double integral and polar coordinates. $$y=3-x$$ $$y^2=4x$$ This is what i figured already: $${r\cos{\theta}+r\sin{\theta}} = 3$$ $$r=0, r=3, \theta=0, \theta=\pi/2$$ ...
5
votes
5answers
83 views

$\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$

Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$ I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting ...
-1
votes
0answers
14 views

b spline, interpolation how many knots required? [closed]

Hi I would like to get help with these questions. How many control points $d_i$ are involved when evaluating a cubic B-spline at a single points. The point are deboor. How many knots are necassary ...
2
votes
1answer
38 views

Double integral - Convert to polar coordinates and find the integration limits by a given domain [closed]

I need help converting to polar coordinates and find the limits of the integrals by this given domain: $$\iint_{D}{} f(x,y)\, dx\, dy$$ $$D= \left\{ (x,y) \mid \dfrac {x^2}{a} \leq y\leq a, -a\leq 0 ...
4
votes
3answers
99 views

Integrating volume of a sphere with a cylinder “drilled” out of it

Unfortunately, I am stuck again on another integration problem. Famous last words, this should be simple. $$ \text{A cylindrical drill with radius 5 is used to bore a hole through}\\\text{the center ...
1
vote
2answers
36 views

partial fraction derivative question

So I have this partial fraction derivative question. I know how to solve it, but for some reason I keep swapping two numbers. Here is the problem: $$\int\frac{3-4x}{x^2+x}= ...
0
votes
1answer
25 views

Integrating a surface bound by a circle

I'm having an issue setting up this problem correctly, regardless of how I seem to do it I end up canceling everything out and getting $0$, which isn't the correct answer. $$ \text{Find the surface ...
1
vote
0answers
22 views

Converting cartesian to polar integral

I feel like I almost have a grasp on regions of integration, I am a bit frustrated that I haven't fully gotten it but because I feel like I'm almost there. In this particular homework problem I have a ...
-3
votes
2answers
35 views

How to find the area under a semicircle using integration? [closed]

How would I go about finding the area under a semicircle? I know that to use integration the formula is $\int_a^b f(x) \mathrm{d}x.$However, when I put this into my graphing calculator it doesn't ...
0
votes
2answers
16 views

Setup region of integration for polar coordinates

I've been working on a homework set for Calc III, right now we're emphasizing double integration and polar integrals. I keep having problems conceptualizing where to actually create my region of ...
1
vote
2answers
32 views

Find total area under infinite curves

My question is finding the total area covered by curves, such as the total area every curve in the following picture covers (from 100 on y axis to 200 on x axis): In my case, the curves are ...
1
vote
1answer
68 views

Stuck on integration question

The curve in the picture shown has equation $y=bx(x-2)$ (a) Find b given that the shaded area is 4 units$^2$ (b) Find the x-coordinate of the point A if the line OA divides the shaded area into ...
0
votes
0answers
25 views

Can you prove that the integral below, with a vectorial field, is zero?

If $\vec{J}(\vec{r})$ is a vector field limited in infinity. Prove that the integral below is zero: \begin{equation} ...
2
votes
0answers
56 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
1
vote
1answer
102 views

Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they might ...
-2
votes
2answers
39 views

How to write the integral $\int_R 5(x+y)\ dy\ dx$ where $R$ is the region bounded by $y=\frac{1}{7}x$, $x=6$ and the $x$-axis?

I have the integral $$\int_R 5(x+y)\ dy\ dx$$ where the region $R$ is bounded by $y=\frac{1}{7}x$, $x=6$ and the $x$-axis. I don't know how to write this problem exactly. Could anyone edit my ...