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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373 views

Sketch the region $R$ and evaluate the double integral $\iint 2y\;\mathrm dA$

Let $R$ be the region in the first quadrant bounded above by the circle $(x-1)^2 + y^2 = 1$ and below by the line $y = x$ . Sketch the region $R$ and evaluate the double integral $\iint 2y \;\mathrm ...
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1answer
183 views

Finding the volume of solid

Use triple integral to find the volume of the solid enclosed between the elliptic cylinder $x^2 + 9y^2 = 9$ and the planes $z = 0$ and $z = x + 3$.
2
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1answer
91 views

Сauchy integral formula [duplicate]

I would like to understand statement of Cauchy integral theorem,which says that Cauchy's integral theorem implies that the line integral of every holomorphic function along a loop vanishes: or ...
2
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2answers
119 views

Finding limits of integration for volume integral

Compute the volume of solid enclosed between the surfaces $x^2+y^2=9$ and $x^2+z^2=9$ What should be the limits of the integrals ? I am getting this z from $-\sqrt{9-x^2}$to $\sqrt{9-x^2}$ x ...
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1answer
54 views

Double Integral Question2

Evaluate $$\int\int_S xdydz+dzdx+xz^2dxdy$$ Where $S$ is the outer side of the part of the sphere $x^2+y^2+z^2=1$ in the first octant. I need some hint to approach this problem.
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1answer
64 views

Triple integral problem

Find volume of the region determined by the inequalities $x>0, y>0,z<8 \text { and } z>x^2+y^2$ using triple integral. i have taken the limits as follows : $z$ varies form $x^2+y^2$ to ...
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0answers
15 views

When an estimator in the Runge-Kutta method (RK4) can not be calculated, what is a suitable estimate?

I'm applying the Runga-Kutta method (RK4) for integrating flow field variables on a finite grid and at some point in time the integration will fail because the integration location is outside the ...
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0answers
139 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
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1answer
61 views

Integration of a matrix over a hypersphere

Can anybody please help me on this one please? $\int_{B({\bf x}_0;R)} \frac{1}{2} ({\bf x} - {\bf x_0})({\bf x} - {\bf x_0})^{T} d{\bf x}$ Here, $B({\bf x}_0;R)$ is a hypersphere(ball?) with radius ...
0
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1answer
51 views

Does the area under the curve remain the same in this variable transformation?

$X$ is a continuous random variable with probability density function $f(X)$. Let $Y = f(X)$. Let $g(.)$ be the pdf of $Y$. Intuitively I think below relation holds true, how to prove it does or ...
9
votes
4answers
486 views

Double integral over a region

Given $f(x,y)=\displaystyle\frac{x^2}{x^2+y^2}$ and $D=\{(x,y) : 0 \leq x \leq 1, x^2 \leq y \leq 2-x^2\}$ i have to solve $\displaystyle\int\displaystyle\int_Df(x,y)dA$. Here's my try: (1) Changing ...
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0answers
145 views

Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
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votes
5answers
124 views

Evaluate $\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$

Evaluate $$\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$$ by any method. In short I am interested in any method that overcomes the lack of convergence of this integral and gives an "number" to it. ...
2
votes
1answer
158 views

Help understanding a concept in multiple integrals

I am learning multiple integrals (Double and Triple Integral) and need help understanding a solution given in the book. In first question, it is asked to find the area lying inside the circle ...
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3answers
323 views

Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}dx$ [duplicate]

Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, dx$$ How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I ...
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1answer
52 views

Can we estimate the lower bound in this way?

This post is aimed to find a lower bound of $\sum_{k=1}^{n}\frac{\cos(kx)}{k}$ for arbitrary $n \geq 1$ ================================= My approach: Let $S_n(x)$ denote the partial sum of the ...
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1answer
194 views

Solving the multiple integral $\int \int_D \sqrt{x^2+y^2+3}dx dy$, D:{$1\le x^2+y^2\le 4, y\le \sqrt3x, y\ge \frac x{\sqrt 3}$}

$\iint_D \sqrt{x^2+y^2+3}\, dx dy$, $D=\left\{(x,y) \in \mathbb{R}^2 \mid 1 \leq x^2+y^2\le 4, y\le \sqrt3x, y\ge \frac x{\sqrt 3} \right\}$. So I've started by drawing two circles, One with $R_1 ...
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1answer
55 views

Parameter integrals - show continuity

One little question concerning the proof of Let $I=[a,b], D=[c,d]$ compact ranges and $f\in C^0(I\times D)$ with values in $\mathbb{R}$ or $\mathbb{C}$. Then $$ F(t):=\int_a^b f(x,t)\, dx ...
2
votes
1answer
725 views

How can I solve this Initial Value Problem using the Runge-Kutta method?

My Problem is this: given Initial Value Problem $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ There is a given Interval of $[1,2]$ and a given step size $h$ of $h=0.1$ After using Euler's and Heun's ...
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1answer
871 views

How can I use the Heun's method to solve this first order Initial Value Problem?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using Heun's method. I have a given Interval of $[1,2]$ and a ...
3
votes
3answers
217 views

For a convex function, the average value lies between $f((a+b)/2)$ and $(f(a) + f(b))/2$

Suppose that $f\in C^2$, $f''(x)\geq 0$ $\,\,\,\forall x \in [a,b]$. I want to show that $$\frac{1}{2}(b-a)(f(a)+f(b))\leq \int_a^bf(t)\,dt\leq (b-a)f\left(\frac{a+b}{2}\right).$$If we divide by ...
3
votes
4answers
446 views

Partial fractions $\int \frac{(3x^2 - 4x + 5)\,dx}{(x-1)(x^2+2)}$

$$\int \frac{(3x^2 - 4x + 5 )\, dx}{(x-1)(x^2+2)}$$ I am going to use undetermined coefficients since it seems straightforward, no wacky matrices or tables to memorize. $$\int \frac{(3x^2 - 4x + 5 ...
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votes
1answer
77 views

Partial fraction $\int \frac{dx}{2x^2 - 3}$

$$\int \frac{dx}{2x^2 - 3}$$ I think I have to factor this but I don't know how to. If I don't know how to is it valid to do $$\int \frac{dx}{2x^2 - 3} \quad =\quad \int \frac {Ax+ B}{2x^2 - ...
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0answers
82 views

Is the antiderivative of $\sec x$ equal to $\ln|\sec x+\tan x| +C$ or $ -2i\cdot \arctan(e^{iz}) + C $?

Quick question here, hopefully you could help me. The anti derivative of $\sec x$ is well known: $\ln|\sec x+\tan x| +C$ Ok, but when I use complex variables, I can write $\sec z$ as: $$ ...
4
votes
2answers
899 views

How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...
0
votes
2answers
519 views

Volume of a rotated region $y = x^2 + 2$

I have no idea what the best method to do this is but I chose the shell method. $y = x^2 + 2$ about $y = -2$ So I convert to f(y) since it rotates on a horizontal axis. $x = \sqrt{y-2}$ I know ...
0
votes
2answers
76 views

Triple integral over a region.

I'm trying to find the volume of $D=\{(x,y,z): \displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} + \displaystyle\frac{z^2}{c^2} \leq 1\}$. I use the change of variables ...
2
votes
2answers
74 views

Solving Bessel integration

What would be the solution of the bessels equation, $$b=k A(t)\int_0^{\infty} J_0 (k \rho) e^ \frac{-\rho^2}{R^2} \rho d \rho$$ Can I sove that by using this formulation? $$c= \int_0^{\infty}j_0(t) ...
1
vote
2answers
59 views

Calculating $a_0$ in Fourier Series

I am using this YouTube video to learn Fourier Series. The question can be clearly seen in the picture. The instructor calculates $a_0$ as the area under the triangle which is fine. Nothing wrong ...
4
votes
2answers
149 views

Does $\int e^\frac 1x \, \mathrm dx$ has a closed form?

How can i solve the integral $\int e^\frac 1x \, \mathrm dx$? I came across this one while trying to do multiple integral on $\int\int_D e^{(\frac xy)} \, \mathrm dA$ where D is the area between ...
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vote
1answer
123 views

Find volume of the body $V = \{ z = \sqrt{6-x^2-y^2}, z = x^2 + y^2 \}$

Find volume of the body $V = \{ z = \sqrt{6-x^2-y^2}, z = x^2 + y^2 \}$ Now what I said is: $$V = \iint_{D} {\sqrt{6-x^2-y^2} - x^2 - y^2 dxdy}$$. But when I wanted to get what $D$ is, I ...
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4answers
6k views

How does knowing a function as even or odd help in integration ??

So, I am learning Fourier Series and it involves integration. I am not too good at integration. Now, the resource I use is videos by Dr. Chris Tisdell. In the ...
1
vote
2answers
130 views

Integral representation of cosh x

On Wolfram math world, there's apparently an integral representation of $\cosh x$ that I'm unfamiliar with. I'm trying to prove it, but I can't figure it out. It goes \begin{equation}\cosh ...
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votes
5answers
264 views

What is the integral of $\int e^x\,\sin x\,\,dx$?

I'm trying to solve the integral of $\left(\int e^x\,\sin x\,\,dx\right)$ (My solution): $\int e^x\sin\left(x\right)\,\,dx=$ $\int \sin\left(x\right) \,e^x\,\,dx=$ $\left(\sin(x)\,\int ...
4
votes
2answers
88 views

Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$

Calculate the volume of $T = \{(x,y,z) \in \mathbb R^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\}$ so I said that the integral we need is $\iint_{D} {x^2 + y^2 dxdy}$. But when I ...
2
votes
1answer
120 views

Volume of a solid of revolution: $y = x^3$, $y = x^{1/3}$, $x \geq 0$ rotated about $y$-axis

I am trying to find the volume: Rotate about $y$. $$y = x^3,\quad y = x^{1/3},\quad x \geq 0$$ Simple enough. $x = y^3 \implies x = y^{\frac{1}{3}}$ $$\pi \cdot \int_0^1 y^{(1/3)^2} - ...
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vote
1answer
68 views

For which a this series converge? $\sum\limits_{n=1}n^{a+1}\int_n^{n+\log(n)}\frac{\arctan(t^a)}{1+t^{2a}}$

For which $a$ $\in$ $\mathbb{R}$ this series converge? $$\sum_{n=1}^\infty{n^{a+1}\int_n^{n+\log(n)}\frac{\arctan(t^a)}{1+t^{2a}}}$$
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vote
2answers
116 views

Volume using disk method $y = 2\sqrt{x} \quad y=x$

$y = 2\sqrt{x}$ $y=x$ about $x=-2$ I know that $y=x$ is on the outside and they meet at 4. I need these in terms of y since I rotate about y. $x = y$ $x = \frac{y^2}{4}$ $$\pi \int_0^4 (y - ...
3
votes
0answers
77 views

Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show ...
1
vote
1answer
59 views

Riemann sums with limits

I just learned about approximation using Riemann Sums, and all that has been taught to us was how to approximate the area under the curve using rectangles. Now, I wanted to try my hand at generalizing ...
0
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3answers
66 views

Taking the derivative of an Integral

I would like to express the derivative of an integral in as elegant a form as possible. However, I am struggling at the moment. I would like to find the derivative $f'(y)$ of the function $f(y) = ...
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0answers
63 views

Complex integral $\frac {1}{1+z}$

I am sure this is quite a trivial problem, but i am stuck and was wondering if anyone could help. I want to solve the integral: $\displaystyle \dfrac{-m-n}{2\pi i}\int \dfrac{dz}{(1+z)}$ for a ...
1
vote
1answer
86 views

Integral $\int_{-\infty}^{\infty} e^{−k^2/(k-k_0)^2}dk$

So I have here the integral $\int_{-\infty}^{\infty} e^{−k^2/(\Delta k)^2}dk$, but then $\Delta k$ is equal to $k-k_0$ so this equation becomes $\int_{-\infty}^{\infty} e^{−k^2/(k-k_0)^2}dk.$ The ...
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votes
1answer
37 views

Concept Of Double Integration

Can someone explain how double integration is equivalent to calculating volume as single integration is calculating area.
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vote
1answer
736 views

Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$.

Again, I am new to volume of bodies and I am struggling with it. Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$. Now from a previous question, I know that I can do it by ...
0
votes
2answers
39 views

How to evaluate base indefinite integrals

I can find a list of known integrals anywhere, but how would I build this list myself? For example how can I prove that $\int x^2 dx = x^3 / 3$? I want to understand the general theory. It would be ...
0
votes
2answers
282 views

Find k such that f is density function

I have the following function: $f_X(x, \theta) = \left\{ \begin{array}{lr} k/x^3 & : x \leq \theta \\ 0 & : x > \theta \end{array} \right.$ and $\theta >0$. I ...
1
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1answer
42 views

Approximate an Integration by a linear formula

I just wonder are there any methods to approximate the following integration by a linear formula ? $$ \int_{x_1}^{x_2} \int_{y_1}^{y_2} f( x,y,w_1,\dots,w_n ) \, dx \, dy \approx \sum\limits_{i = ...
1
vote
1answer
43 views

Problem with understanding integration by parts method?

First of all (before I get started with integration by parts method to solve integrals) I need to know what is the meaning of all terms in integration rules like $a^x,\,e^x,\,\ln\left(x\right),\,$ ...
4
votes
3answers
284 views

Evaluating $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ using polar coordinates

Use polar coordinates to evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ I understand that we need to change $x^2+y^2$ to $r^2$ and then we get $\int_0^1 \int_0^{\sqrt{1-x^2}} ...