8
votes
1answer
58 views

Changing the order of integration without sketching?

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...
1
vote
1answer
39 views

Inner product of functions as integration

I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties. My ...
4
votes
2answers
83 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
1
vote
2answers
65 views

How wrong is it? - A “proof” of the FTC that I came up with in high school by hand-waving.

In high school calculus, I was first taught that the area under a curve $f(x)$ between $x=a$ and $x=b$ is given by: $$ A = \lim_{\delta x \rightarrow 0} \sum \limits_{a}^{b} f(x) \delta x $$ Then ...
6
votes
4answers
138 views

Evaluate $\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$

I need to evaluate the integral: $$\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$$ for $0<p<1$. Unfortunately I do not know where to begin. I tried integration by parts but got nowhere ...
3
votes
2answers
40 views

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as ...
2
votes
2answers
35 views

Question in relation to Fundamental Theorem of Calculus

I obtain two answers, one is $\dfrac1{\sqrt{1+x^6}}$ and another one is $\dfrac{2x}{\sqrt{1+x^{12}}}$ by using Fundamental Theorem of Calculus, but I am not so sure. Would anyone help me?
1
vote
1answer
54 views

Washer method and shell method

(1) Sketch the region enclosed between the curve $y=sin^2x$ and the straight line $y =2x/π$ (2) Find the volume of the solid $S$ obtained by revolving the region in part (1) about the $y$-axis by ...
2
votes
2answers
39 views

Evaluating expression at infinity

How do I evaluate something like: $$xe^{-(x-\theta)}\text{ from }x = \theta\text{ to }x=\infty?$$ This came up in an integration I tried to do, and I realize it's a very basic question. But I am ...
2
votes
2answers
66 views

Evaluating integral limit in two ways gives different limits

Problem Show that $$\lim\limits_{h \rightarrow 0^{+}} \int_{-1}^{1} \frac{h}{h^{2}+x^{2}} \, dx = \pi.$$ I can do this by evaluating the integral directly and showing that it is equal to ...
3
votes
3answers
79 views

Integrating a function with an infinite number of discontinuities

I would appreciate some help with the following exercise: Let $$f(x)=\begin{cases} 1 & \text{if}\ x= 1/n\ \text{for some}\ n \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$ Show that ...
1
vote
4answers
88 views

Does $\int_1^\infty \frac{\log x}{x^{3}} \sin x \,dx $ exist?

I want to determine if the following indefinite integral exists: $$\int_{1}^{\infty} \frac{\log x}{x^{3}} \sin x dx.$$ I tried to solve the integral then calculate the limit $$ \lim_{\lambda \to ...
4
votes
1answer
139 views

indefinite integral $\int\sin\sqrt[3]{x}~dx$

I want to determinate the integral $\int\sin\sqrt[3]{x}~dx$ . I tried to use integration by partitions and integration by substitution but I came to no result. I know the result which is shown here ...
2
votes
0answers
30 views

Calculus: Reduction formula

For this question, I can find out $I3$, but I have no idea how to find the reduction formula. Please advise me.
2
votes
1answer
62 views

Calculating joint MGF

This is an end-of-chapter question from a Korean textbook, and unfortunately it only has solutions to the even-numbered q's, so I'm seeking for some hints or tips to work out this particular joint ...
1
vote
1answer
35 views

How to evaluate: $\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy$

How can I compute the integral: $$\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy\quad\text{where}\ z \in (0,1) $$ Had it not been for $z$ , the integral would look like an incomplete beta function but what ...
0
votes
2answers
51 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
0
votes
1answer
52 views

Most powerful size $\alpha$ test

Someone can help me to check this answer? How to find the Most Powerful Test size $\alpha$ and Power of Test, Since I have $H_0 : X \thicksim f_{\theta 0}= (1/\sqrt(2\pi) \exp^{(-x^2/2)}$ and $H_1 : ...
1
vote
1answer
91 views

Leibniz rule in a Double Integral

I have been trying to evaluate the following double integral: $$\frac{\partial}{\partial \theta_1 \partial \theta_2} \int_{\theta_1-\theta_2}^{\theta_1+\theta_2} \int_{\theta_1 -\theta_2}^{x} u(y,x) ...
2
votes
1answer
84 views

Evaluate $\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$

How can I evaluate the following double integral: $$\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$$ If we didn't have the restrictions $x<u, y<v$ polar coordinates would have ...
0
votes
0answers
38 views

The part I dont understand while calculating contour integral.

Question is the following; $$\int_{0}^{\infty}[x^{m-1}/(1+x^n)]dx$$ for $m,n=1,2,\dots$ and $n>m>0$ Solution: its poles were found as this $$a_k=e^{i(2k+1)\pi/n}$$ for $k=0,1,...(n-1)$ And ...
1
vote
3answers
74 views

Evaluation of the integral.

$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$ The integration is taken along the imaginary axis, an integer ...
3
votes
2answers
205 views

Good books to learn Riemann integration

I am looking for a good text book to learn Riemann integration. Please suggest books with theories and proofs comprehensively explained.
1
vote
2answers
60 views

Integration by Parts confusion

I am using this video to learn Laplace Transform. The example used is a fairly basic one: $$ \int_{0}^{\infty}t.e^{-st}dt $$ Simple enough, you need to integrate by ...
2
votes
2answers
142 views

How to solve this “simple” ordinary differential equation?

I am trying to learn more about calculus by myself, in order to be able to use dynamical systems analysis methods. In a book example, I have to find $f(t)$ from this: ...
2
votes
2answers
86 views

Assume that $f(x)\ge0$ if $x$ is a point in $\bf{I}$ with a rational component. Prove that $\int_{\bf{I}}f\ge0$.

Let I be a generalized rectangle in $\Bbb R^n$ and suppose the function $f:\bf{I}\to\Bbb R$ is Riemann integrable. Assume that $f(x)\ge0$ if $x$ is a point in $\bf{I}$ with a rational component. Prove ...
3
votes
1answer
214 views

A 'complicated' integral: $ \int \limits_{-\infty}^{\infty}\frac{\sin(x)}{x}$ [duplicate]

I am calculating an integral $\displaystyle \int \limits_{-\infty}^{\infty}\dfrac{\sin(x)}{x}$ and I dont seem to be getting an answer. When I integrate by parts twice, I get: $$\displaystyle \int ...
0
votes
1answer
74 views

Learning Fourier Integral

I am learning Fourier Integral for real numbers. I downloaded a presentation of some university. I am summing up what I know and hopefully someone will correct me where I am wrong. Fourier ...
0
votes
1answer
55 views

Contour integration of complex number confuses me, still.

Given $f(z) = (x^2+y)+i(xy)$ and we integrate it using the Parabola Contour. For a parabola, $\gamma(t) = t + it^2$. So, $f(\gamma(t)) = 2t^2 + it^3$. What was ...
0
votes
1answer
135 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
0
votes
1answer
32 views

Am I understanding this integration right?

This is the snippet of a problem from this PDF here. What I dont understand is why they retain the $Sin$ part for evaluation after integration when all that it is going to evaluate to is 0. If I ...
0
votes
1answer
61 views

Find the integral of $\overline{z}$

Question: Find $\int\overline{z}$, when the contour is a parabola. Interval is from 0 to 1. My Attempt: $z = x + iy \Rightarrow \overline{z} = x - iy$ $f(z) = x - iy$ Since the contour is a ...
0
votes
2answers
142 views

Difficulty in understanding integrals of complex numbers

I understand what integration of real numbers is. I know how the definition of it is made. I have trouble in understanding how it works for complex numbers. I am referring to the notes here: ...
2
votes
1answer
102 views

Can somebody provide an explanation to the formula of a one elementary integral?

Here is the formula: $$ \int{\frac{dx}{x}} = \ln{|x|} + C $$ In my textbook it is given without proof, so I have a little confusion here. From the definition of integral this equality must be true: ...
0
votes
1answer
124 views

Integrate $\int \frac{1}{x^3}e^{-x^2}dx$

$$\int \frac{1}{x^3}e^{-x^2}dx$$ What I did? put $1/x^2 = t; $ then $\int \frac{1}{x^3}e^{-x^2}dx$ will trasform into $\frac{-1}{2}\int e^{-1/t}dt$ I don't understand how to proceed there after. ...
3
votes
0answers
230 views

Are Specific Facts about the Riemann Integral Logically Required?

This question is somewhat in the spirit of this one in that I am trying to understand the most efficient path to the major integral theorems (Fubini, change of variables, etc). My question is this: ...