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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How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
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22 views

A condition for a function to be complete measurable

I was doing this exercise in Real Analysis of Folland and got stuck on this problem. I get no clue about how to define these 2 functions series, so I hope some one can help me solve this. I really ...
3
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1answer
88 views

Integration and theorems on continuous functions

$f(x)$ is positive and continuous function on $\mathbb{R}$ and, moreover, $\int_{-\infty}^{+\infty}f(x)dx=1$. $\alpha\in(0;1)$ and $[a;b]$ is the interval having a minimum length such that the ...
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3answers
100 views

integral computation $\int_{-\infty}^{\infty} \frac{1}{(1+x+x^2)^2} dx $

Compute the following integral: $\int_{-\infty}^{\infty} \frac{1}{(1+x+x^2)^2} dx$. Can some one give me some hints on how to do this? I tried writing $(1+x+x^2)=f(x)$ and then multiplying and ...
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1answer
56 views

Integral with the incomplete upper gamma function

Can anyone help me integrate this? $$\int_0^1 \frac{1}{x^{1/p}} \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{m/n-1} \Gamma\;\left(A, \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{1/n}\right) ...
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1answer
275 views

A rod with density $\delta(x) = x^2+2x$ lies on the $x$-axis between $x= 0$ and $x= 2$. Find the mass and center of mass of the rod.

A rod with density $\delta(x) = x^2 + 2x$ lies on the x-axis between $x= 0$ and $x= 2$. Find the mass and center of mass of the rod. I found mass by integrating $\int_0^2 \left(x^2+2x\right)$ and got ...
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0answers
114 views

Convolution of piecewise function

I would like to compute the convolution of piece wise function Following is the piecewise function $$ C_a(t) = \begin{cases}0& t\leq t_d\\ ...
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2answers
247 views

Find the volume bounded by a cylinder.

Find the volume bounded by the cylinder $x^2 + y^2=1$ and the planes $y=z , x=0 ,z=0$ in the first octant. How do I go about doing this?
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1answer
25 views

U-Substitution for Volume Integral

I can't figure out where I'm going wrong with the following integral: Given: $$ V = \pi\int_1^2 (2-\frac{x}{2})^2 dx$$ Substitutions: $$ u = 2-\frac{x}{2} $$ $$ du = -\frac{1}{2} dx$$ $$ (-2)du = ...
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12 views

Finding new state via Euler integration

I have a state; call it X = [x,y, $\theta$]. It's a pose. I know my linear velocity (v) and my angular velocity ($\omega$) (or at least decent approximations of them). I've been asked to find the new ...
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0answers
49 views

Prove that if two functions, f and g differ at a finite number of points then the lower integrals are the same

question: take two functions $f,g \to [a,b]$ to be bounded and suppose that $f(x) \not = g(x)$ at a finite number of points. Prove that $\underline{\int_a^b} f(x) \ dx = \underline{\int_a^b} g(x) \ dx ...
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2answers
113 views

Show that $\int_{a}^{b} |f|^2\le\frac{(b-a)^2}{2}\int_{a}^{b} |f'|^2$

Let $f$ be an differentiable function. Show that $$\int_{a}^{b} |f|^2\le\frac{(b-a)^2}{2}\int_{a}^{b} |f'|^2$$ Can you give me any Hint please (why not, $(b-a)^2$, where does the $\frac{1}{2}$ ...
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0answers
140 views

How to calculate an integral with a step function squared?

how can this integral be calculated: $\int_{-\infty}^{\infty} [e^{-2mx}\:\theta^2(x) + 2\theta(x)\theta(-x) + e^{-2mx}\:\theta^2(-x)] dx $ where $\theta(x)$ is the unit step function with its ...
2
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2answers
73 views

Did I solve this integral correctly? (trig substitution)

I'm having trouble with trig substitution. This is what I've done so far, but I'm not sure if I did everything right. This is the integral: $$\int \frac{x^2}{(1+x^2)^\frac{3}{2}}$$ and my ...
2
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1answer
29 views

Integral of $|f|$ outside a compact set

Let $G$ be a locally compact group. Given $f\in L^1(G)$ and $\epsilon>0$, how to show that there is a compact set $K\subset G$ such that $\int_{G\setminus K}|f|<\epsilon$?
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192 views

Dual Integral Problems: $\int\left(\pi^{\sin^2x}+e^{\sin^2x}\right)^2\cos2x~dx$ and $ \int\frac{\sin x}{1+\cos x+e^x}dx$

Recently, I saw these two calculus problems showed up on internet. I tried to answer those problems but I couldn't. I've already used WolframAlpha to figure it out, but the result was nothing. These ...
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1answer
67 views

A nice formula for an integral of an increasing step function

Suppose $\{a_n\}_{n=0}^\infty$ is a monotone increasing sequence with $a_0=0$. Define the following function $f_{a_n}:[0,\infty)\to\mathbb{R}$: $$f_{a_n}(x)=\begin{cases}k&a_k\le ...
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0answers
55 views

Calculate the area bounded by D in the first Quadrant where $D = \{(x,y): x \le y \le 3x\, , \, 1 \le x \le 2\}$

Calculate the area of $D$ where $D$ placed on the first quadrant and bounded by: $x \le y \le 3x$ and $1 \le xy \le 2$. Hint: change variables. My try: $x \le y \le 3x \implies 1 ...
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20 views

Using integrating techniques to get T(t) on both sides

So I have to solve a wave equation with the following boundary condition. m$\frac{\partial ^2u}{\partial t^2}$(L,t)=-AE$\frac{\partial u}{\partial x}$(L,t) so I have rewritten in this form ...
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2answers
177 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
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1answer
78 views

Calculate $\iint_D 3dxdy$ where $D = \{(x, y) : (x+y)^2 + (2x - y)^2 \le 4 \}$

Calculate $\displaystyle\iint_D 3dxdy$ where $D = \{(x, y) : (x+y)^2 + (2x - y)^2 \le 4 \}$. I tried to solve this can I failed. Can you please give me some hints?
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1answer
442 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
4
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1answer
206 views

“Ito-Riemann” integration

Is there a real-valued function on a closed bounded nonempty interval that is not Riemann integrable, but is "Ito-Riemann" integrable, that is: if the sampling point is always the left-end point of ...
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63 views

Construction of Lebesgue integral

I have a couple of questions regarding the construction of the Lebesgue integral. I am looking at one construction based on simple functions that reads: Definition: A measurable function $f : ...
3
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0answers
98 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
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1answer
48 views

Limits of integration in a double integral after changing the coordinates

I do not understand why low limit is $u=-rw$, and upper limit is $w$. It is from paper Appendix C of On the joint statistics of stable random processes by K. I. Hopcraft and E Jakeman.
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42 views

fourier series and correlation coefficients question?

We have the signal in the figure. I must do the trigonometric fourier series of the signal and also the exponential fourier series.Also,find the correlation coefficients between $f(t)$ and $e^{3t}$. ...
2
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1answer
99 views

Definite integral involving arctan and tan

I was solving a problem posed on Moldavian National Mathematical Olympiad for 12th grade in 2012. The question was the following: Problem. Let $f:\mathbb{R}\rightarrow\mathbb{R}$, such that ...
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2answers
58 views

How to solve the ordinary differential equation $\frac{dv}{v(1 + v^2/c^2)} = \tau_0 \,dt$

Could you assist me to solve this problem? $$ \frac{dv}{v(1+ v^2/c^2)} = \tau_0dt. \qquad (\tau_0 = 1.5)$$
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3answers
54 views

Finding the antiderivative

Let $$ \int {\frac{{1 + \frac{1}{{{x^2}}}}}{{{x^2} - 1 + \frac{1}{{{x^2}}}}}} dx $$ I've been told use the substitution: $t = {{x-1} \over x}$. But how to apply it on the integral?
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0answers
25 views

Measure of the intersection of hypersphere and hyperbox

I have an origo-centric unit hypersphere $S^{d-1}=\{\mathbf{x}\in\mathbb{R}^d|\;\|\mathbf{x}\|=1\}$ and a hypercube of side length $l$ at point $\mathbf{x}_0$ (maybe easiest if it is the "lower-left" ...
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1answer
56 views

Double integral cos(y)

I have no idea how to calculate this kind of double integral. $\int_D \cos(y) ~dA$ where $D=\{ 0 \leq x \leq 2\pi,~ |y|\leq x \}$. Any help with this? (added from now deleted answer) Okay, I ...
2
votes
1answer
108 views

Prove that $\int_{E}f =\lim \int_{E}f_{n}$

I'm doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can't come to the conclusion. Can anyone help me. I really appreciate. Consider a ...
2
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0answers
31 views

Proof of a Proposition

I am having trouble trying to proof a proposition that appears in a paper. It begins with ...
0
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2answers
34 views

integral of parametric equation

Let g(x) = $\int_{-1}^{2x+4} e^{-t^2} dt$ Find the value of g'(-2). The answer is 2, but I don't know how to get there. I thought that the derivative of the integral should just be $e^{-t^2}$ and ...
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2answers
73 views

Question about integral is equal to zero

Suppose we have the following equation $$ \int_0^\infty {f(x,r)g(x)dx} = 0 \quad {\rm for \, all}\, r\in \mathbb{R} $$ where the function $g(x)$ does not depend on $r$, while $f(x,r)$ is function of ...
2
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1answer
52 views

integrating something over dy

How can I integrate the following: $$ \int \frac{t+1}{dt}=\int\frac{-y^2}{dy} $$ would I integrate normally and inverse it? like: $$\int\frac{1}{dy} = \frac{1}{y} + c $$ Thanks.
2
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2answers
144 views

Derivative w.r.t. limits of integration of double definite integral (Leibniz' rule)

How does Leibniz' rule work with a double definite integral, when the limits of integration of the inside integral depend on the variable in the outside integral? For example, how do we calculate the ...
2
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1answer
48 views

How to simplify path integral?

I am trying to integrate a function, $f(x,y)$, over the straight line path connecting $(0,k)$ to $(k,0)$ in the x-y plane, where $k>0$ (the diagonal part of the boundary of a simplex in ...
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82 views

How I cut my orange - spherical volume integral

I cut my orange in six eatable pieces, following some rules. My orange is a perfect sphere, and there is a cylindrical volume down through my orange, that is not eatable. In the diagram, the orange ...
3
votes
1answer
97 views

Integration by parts seems to be infinite, but answer is simple

I have to integrate this: $$ \int_{-1}^0 \int_{-2x-2}^{2x+2} x^2 y^2 + \sin(xy) e^{x^2} y^2 \;\operatorname d\!y \operatorname d\!x $$ I started integrating by parts, $$ \int_{-1}^0 e^{x^2} ...
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0answers
41 views

Efficient approximation of derivatives of an integral

Suppose $ \phi(z) $ is the probit function (http://en.wikipedia.org/wiki/Probit). And $$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ...
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0answers
40 views

Problems with the integration theorem of stokes.

i got problems with the next question: Let $w(x,y,z) = \left(zx+z^2y+x.z^3xy+y,z^4x^2\right)$ and let $P = P_1 \cup P_2$. $P_1 = \{(x,y,z) \in \mathbb{R}^3|x^2+y^2+(z-1)^2=1, z \geq 1 \}$ $P_2 = ...
5
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2answers
101 views

Integral $\int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} \operatorname d\!x \operatorname d\!y\,$

I have to evaluate this integral: $$ \int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} \operatorname d\!x \operatorname d\!y\, $$ I have no idea what to do with $\;{e}^{x^7}$. I have even tried $\int{e}^{x^7} ...
5
votes
2answers
5k views

U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong?

Here is my work, witth the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How? $$ \int \frac{1}{1+e^z}dz = \int\frac{1}{e^z(\frac{1}{e^z} + 1)}dz ...
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2answers
139 views

Use the Shell Method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis?

$y= 8x^2$, $y=8 \sqrt {x}$ I know the limits are $0$ and $8$. But do I change the function so it's "$x=$" or leave it as it is?
2
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1answer
52 views

How to calculate this exponential integral ?

How to calculate this integral? : $$∫(1+a/x)^{x}dx$$ here $a∈ℝ^{∗}$.
0
votes
2answers
53 views

Set up an integral for the length of the curve.

$$x= y^{1/30},\; 0 ≤ y ≤ 2$$ I know the formula we use is $\sqrt{1 +f'(x)^2} dx. $ But now do I switch the function so it's "y="?
1
vote
1answer
71 views

Use the Shell Method to find the volume by revolving around the x-axis.

The functions are $x=\frac{y^2}{2}$, $x=2$, and $y=2$. I graphed it and it looks like the intersection points are $(2,2)$ and $(0,2)$. But I don't know how to set up the integral.
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3answers
75 views

How to solve the integration? [closed]

Solve the Integral : $$\int_0^\infty \lfloor x\rfloor e^{-x}dx, \text{ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.}$$