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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3answers
48 views

Area of a rectangle within a curve

The cargo space of a bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. It is shaped like a parabola with equation ${{1\over 4}x^2, - 6 \le ...
1
vote
1answer
20 views

Computing a line integral where the curve is in polar coordinates

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(y,x)$ and $C$ is the curve given by $r=1+\theta$ for $\theta \in [0,2\pi]$ My Attempt Am I correct in saying that $F$ is a conservative vector field ...
1
vote
1answer
86 views

How to show the following function is Riemann Integrable

We have not covered and thus it is not valid to use ideas such as Lebesgue integration, measure, etc. I was given a hint to use either squeeze theorem, or the criterion about if Riemann integrable ...
-1
votes
2answers
53 views

Evaluate the following integral?

Evaluate the following integral. $\int\frac1{(x+a)(x+b)}~dx$ ${a}\neq{b}$ I do not know where to being solving this integral.
3
votes
3answers
58 views

Find $\lim_{n \rightarrow \infty} \int_0^n (1+ \frac{x}{n})^{n+1} \exp(-2x) \, dx$

Find: $$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$ The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So ...
2
votes
2answers
67 views

Compute integral: $\int_0^{+\infty}\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-\frac{1}{2} (x^2+y^2)}dx dy $

I would like to resolve this exercise: Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$. I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ ...
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votes
1answer
45 views

Integration of an equation containing Legendre Polynomials

Consider the Integration $\int_{-1}^{1}x^{2}P_{n+1}(x)P_{n-1}(x)dx$ where, $P_{n+1}(x) ,P_{n-1}(x)$ are Legendre Polynomials Applying integration by parts,we get ...
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2answers
41 views

Evaluate the inverse trigonometric integral

Evaluate the integral:$\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx$ On applying the property $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ I dont seem to reach any where
5
votes
2answers
148 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
1
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1answer
47 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
0
votes
0answers
54 views

Surface Integral over Ellipse

Let $$E=\left\{(x_1,x_2)~\middle|~\frac{x_1^2}{ a^2}+\frac{x_2^2}{b^2}=1\right\}=\left\{X(\theta)=(a\cos(\theta),b\sin(\theta))\,\middle|\,0\leq \theta\leq 2\pi\right\},$$ be an ellipse. Let ...
2
votes
1answer
63 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
0
votes
1answer
47 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
1
vote
1answer
27 views

Evaluating the integral of a sine function

I am having some trouble with part (b) and part (c) of this: (b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos ...
1
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2answers
691 views

Explain why graph of f lies below the $x$-axis in interval $[4\pi/9,5\pi/9]$

$f(x)=(x+1)\sin(3x)$ Explain why the graph of f lies below the $x$-axis for values of $x$ in the interval $[4\pi/9, 5\pi/9]$ From what i know/understand I'd have to look at the function in two ...
1
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2answers
139 views

Cross Product for Biot-Savart Derivation of Current Loop

Biot-Savart's law can be used to determine the magnetic field produced by a figure at a point. Introductory physics texts integrate $dB$ to obtain $B$ where $$dB = \frac{I\mu_{0}}{4\pi r^2} dl ...
0
votes
5answers
64 views

find the area of a kite with integration

A stunt kite has the shape in the diagram below: How can I find the area using calculus integration. Can anyone help me start this question, I am not looking for the full answer. I assume I only ...
1
vote
1answer
51 views

How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$? To start ...
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votes
0answers
24 views

Definition of integrability for sequences

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
1
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1answer
40 views

Graphical Convolution

For the first part of the above problem, I copied an example from my book and I got the answer to be $$t(t-1)+t(t-2)=t^2-3t$$ considering that the integral is the sum of the area of the rectangles ...
0
votes
1answer
29 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
1
vote
2answers
79 views

Calculate $\int_{0}^{1}\left \{ \sqrt{1-x^2}+2 \right \}^2 dx$

I couldn't find any suitable substitution for this integral and hence I couldn't solve it. $$\int_{0}^{1}\left \{ \sqrt{1-x^2}+2 \right \}^2 dx$$
7
votes
1answer
2k views

A function of two cumulative probability distributions with same first 2 moments

Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment (hence ...
2
votes
1answer
2k views

Calculating an integral derived from the convolution of two Fourier transforms

Let $\sigma>0$ , $1<\alpha\leq 2$, and $-1\leq \beta \leq 1$. I am looking for a closed-form solution (or something near) for the following integral. $$\frac{1}{2 \pi } \text{PV}\int_{-\infty ...
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votes
0answers
37 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
1
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2answers
539 views

Any tips to solve this integral : $I_1 = \int \ln(x^2)e^{\sin(x)}\sin(x^{\cos(x)}) dx$

Background: I was making new expressions to see whether I could efficiently find their derivatives... After having done that, I've started trying to integrate most of them; obviously most of them ...
1
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2answers
31 views

Find volume of these solids using integration

a) The $(x>0, y< -1)$ region of the curve $y= -\frac{1}{x}$ rotated about the $y$-axis. The instructions say that one should use the formula: $V = \int 2πxf(x) dx$ I used another method and ...
0
votes
0answers
42 views

Finding the integral of a 1/variable*radical function

I'm trying to find the integral of $$\int\frac{1}{x* (\sqrt{4x^4 - 9})}$$ Attempt: I assumed that the integral would be some sort of inverse trigonometric function. Because of this, I did the ...
0
votes
1answer
22 views

Finding volume of a solid of revolution

I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis. If I'm correct, this volume can be calculated by: Evaluating ...
0
votes
1answer
53 views

Having Troubles With This Integration Problem

The question I'm having troubles with is as follows: Evaluate $\int_{-r}^r\sqrt{r^2-t^2}\,dt$ (Hint: substitute $t=r\sin x$) So, immediately I did $dt=r\cos x\,dx$ and substitute it all in... ...
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0answers
24 views

Integral of combination of power, exponential, and kummer hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(x-b)^{2}}\, M(-\alpha,-\beta,\lambda x) \end{equation} $\alpha > 0$ and $\beta > 0$ are ...
0
votes
1answer
28 views

Parametrize the given curve and compute the integral (complex numbers)

The integral I have to evaluate is $\int_Czdz$, where $C$ is the line from 0 to $1+i$, and then from $1+i$ to 2. My work: $z_1(t)=(1+i)t$ and $z_2(t)=(t+1)+i(1-t)=t(i-1)+(1+i)$, $t\in[0,1]$. ...
0
votes
2answers
54 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
1
vote
4answers
113 views

How to integrate $\int \frac{dx}{x^2 \sqrt{x-1}}$?

I need to integrate$$\int \dfrac{dx}{x^2 \sqrt{x-1}}.$$ I've tried everything from substitutions ($\sqrt{x-1}=u$) to integration by parts but I cannot get anywhere. Please help.
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votes
2answers
49 views

What is the value of integral? [on hold]

Let $y(t)$ be a continuous function on $[0,\infty)$. If $$ y(t)= t\left(1-4 \int^t_0 y(x) dx\right) +4 \int^t_0 xy(x) dx$$ then what is the value of $\int^{\frac{\pi}{2}}_0 y(t) dt\,$?
1
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2answers
29 views

how could calculate $ \int_{C} \frac{1}{\sin(z)} \, dz $ when $C=C(0,1)$

i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
0
votes
0answers
32 views

Leibniz integral rule definition

https://en.wikipedia.org/wiki/Leibniz_integral_rule If we have an integral $$\int_{y_0}^{y_1} f(x, y) \,\mathrm{d}y$$ then for $x$ in $(x_0, x_1)$ the derivative of this integral is thus ...
0
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0answers
15 views

Relationship between Lippmann-Schwinger integrals of different dimensions

Define $G_n (\mathbf{x},\mathbf{x}')$ as $$ G_n (\mathbf{x},\mathbf{x}') = \lim_{\epsilon \to 0^{+}} \left[\dfrac{1}{(2 \pi \hbar)^{n}} \int_{\mathbb{R}^{n}} \mathrm{d}^{n}\mathbf{p} \dfrac{e^{i ...
0
votes
0answers
78 views

I am a Math Hobbyist. I have made some simple discoveries in Math. How do I share it with the Math community out there? [closed]

I am a Computer Engineering graduate and have taken many courses in Math of course. While I was in the University, I got myself lost in the world of mathematics and I discovered stuff that I felt ...
2
votes
2answers
30 views

How could I calculate $\int_{C} ze^{\frac{1}{z-1}}$ when $C=C(1,\frac{1}{2})$

I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$ I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue. What I did: ...
0
votes
1answer
41 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
1
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1answer
38 views
+50

Complicated surface integral/line integral.

Problem Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$ And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$ For $F=(x^2y,3x^3z,yz^3)$, and ...
2
votes
1answer
73 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
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votes
0answers
14 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
1
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1answer
23 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
1
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4answers
10k views

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer)

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer) I saw no example of this problem anywhere.. I saw an ...
0
votes
1answer
17 views

Surface Integral of $3z^2 d\sigma$

Let $S$ be the bounded surface of the cylinder $x^2+y^2=1$ cut by the planes $z=0$ and $z=1+x$ Then how to show that the value of the surface integral $∬3z^2 d \sigma $ over $S$ is equal to ...
1
vote
1answer
45 views

$E=\left\{(x_1,x_2,x_3,x_4):\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$, what is $\int_E e^{-\langle x,t\rangle} \, dx$?

I'm learning some real analysis and encountered the following question: Let $E=\left\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$. for which values of ...
-1
votes
1answer
54 views

Prove that a function is Riemann integrable directly, using $\epsilon-P$

I know there already are questions like these, but I still don't understand how to prove it. Question: Prove that $f$ is Riemann integrable on $[0,1]$ if $$f(x) = \begin{cases} x^2 \sin (1/x) ...
0
votes
0answers
52 views

Integration of $\frac{x^2}{2\left(e^x+1\right)}$

Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...