Integral calculus Definition and 156 Threads

Say we have a function of the form f(x)=\bigg\lbrace\begin{matrix}c \qquad\quad\text{x=0}\\ 0\quad\text{elsewhere}\end{matrix} If we then integrate this over all space i.e. \int_{-\infty}^{\infty}f(x)dx Why does the result equal zero?

Kindly can someone tell me how can i integrate and plot the following formula as a function of time. while theta is changing from 1 to 20. also attached .docx file for convenience. Any kind of help or advice will be highly appreciable.

Look, I was wondering if substituting the variable more than once is valid and hence the definite integral intervals change this way. Consider the following integral (I'm working for finding the volume of a solid of revolution): *\pi \int_{-3}^{5}3^{2}-(\sqrt{\frac{y+3}{2}}+1)^2dy Personally I...

Homework Statement ( V1.4 A.4 C1 - (1/V) ) dV = dA / A C1 is a constant, V=f(x), A=.25*pi*(.0222 - x2) I'm trying to simplify the equation into a form with no integrals or derivatives, so I can put it into MATLAB to spit out an expression for V as a function of x. Or is it possible to put all...

Hi, I have been trying to find an integral ## \int_{-\infty}^{+\infty} \frac{ e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2} dx ##. I initially posted this question in the complex analysis forum since I felt it might be done using contour integration. However now I realize it might not be the best way...

Homework Statement (hebrew) : f(x) a continuous function. proof the following Homework Equations I guess rules of limits and integrals The Attempt at a Solution I've tried several approaches: taking ln() of both sides and using L'Hospitale Rule. Thought about using integral reduction...

Homework Statement Evaluate the integral of (x+1)5^(x+1)^2 Homework EquationsThe Attempt at a Solution I set my u=(x+1) making du=1dx. This makes it u*5^u^2. I integrated the first u to be ((x+1)^2/2) however I don't know what to do with the 5^u^2

Hi there, I have a question I'm stuck on. It is: Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analysed from a photograph. Suppose that in a spherical cluster of radius R the...

I understand that the fundamental theorem of calculus is essentially the statement that the derivative of the anti-derivative F evaluated at x\in (a,b) is equal to the value of the primitive function (integrand) f evaluated at x\in (a,b), i.e. F'(x)=f(x). However, can one imply directly from...

I've been reading Wald's book on general relativity and in one of the questions at the end of chapter 2 he gives a hint which says to make use the following integral identity (for a smooth function in): F(x)=F(a)+\int_{0}^{1}F'(t(x-a)+a)dt Is this result true simply because...

In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence: \begin{align}\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot \mathbf{J} ) d^3 x\end{align} He calls this change "integration by parts". If this...

Homework Statement Find all values of a such that this set is a type II region (i.e. the bounds of x can be represented as functions of y, while the bounds of y are constant valued) -1<y<0, Y<x<-y (union) 0<y<1, -y+a<x<y+a Homework EquationsThe Attempt at a Solution I arrived at a being any...

Homework Statement Find whether the integral is convergent or not, and evaluate if convergent. Homework Equations integral 1/sqrt(x^4+x^2+1) from 1 to infinity The Attempt at a Solution 1/sqrt(x^4+x^2+1)<1/sqrt(x^4) 1/sqrt(x^4)=1/x^2 which is convergent for 1 to infinity and is 1 therefore...

Recently I started seeing integral calculus and right now we are covering the topic of the antiderivative. At first sign it was not very difficult, until we started seeing integral variable substitution. The problem starts right here: Let's suppose that we have a function like this: \int...

hi , i need some help in this integral e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinityI tried differentiating e^(-bx^2) twice and it came up weird , is there any other way to do it ?

Hi! I have some problems with the integral $$\iiint_T\sqrt{x^2+y^2}z^4e^{z^4}dx\ dy\ dz$$ where $$T=\{(x,y,z)\in\mathbb{R}^3:\sqrt{x^2+y^2}\leq z\leq 1\}$$ I have tried to change it to spherical and cylindrical coordinates but... nothing Can someone help me? Thanks

This is a problem I've been working on... I did some things wrong, and syntactically incorrectly, and just messy. Could you help me out? Tell me how I can write this more syntactically correctly, and cleanly, please? Also, I realize that some might believe this would be more appropriately put...

Am I doing good on this problem?

Author: Richard Courant Title: Differential and Integral Calculus Amazon Link: https://www.amazon.com/dp/4871878384/?tag=pfamazon01-20 https://www.amazon.com/dp/487187835X/?tag=pfamazon01-20 Prerequisities: Table of Contents of Volume 1: Introductory Remarks Introduction The Continuum of...

I'm just a high school senior who noticed that the derivative has a general formula that we all know is, \frac{f(x+h)-f(x)}{h} but that there is no general formula (at least I haven't heard of it yet) for the integral of a function. I know I cannot simply just take the inverse of the difference...

Reading the Integral Calculus tutorial, I felt like contributing. Do with it what you wish... Proposition: You can evaluate areas exactly using Integrals. Knowing that an integral is an antiderivative, and that derivatives are RATES, it seems odd that going in reverse would give geometric...

1. Prerequisites In order to study integral calculus, you must have a firm grasp of differential calculus. You can go to http://mathhelpboards.com/calculus-10/differential-calculus-tutorial-1393.html for my differential calculus tutorial, although, of course, there are many books and other...

Homework Statement A generated AC voltage has a value given by V = 4cos2θ. You will need to use the identity cos^2⁡θ = 1/2(1+cos2θ) a) Find the mean value of the voltage over a full cycle (0 ≤ θ ≤ 2π) using integral calculus b) Find the r.m.s. value of the voltage over a full...

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Homework Statement Evaluate \iiint_\textrm{V} |xyz|dxdydz where V = \{(x,y,z) \in ℝ^3:\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ≤ 1\}Homework Equations Change of Variables Theorem: \int_\textrm{ψ(u)} f(x)dx = \int_\textrm{K} f(\Psi(u))|detD\Psi(u)|du Examples: 1) For a ball of...

Integral calculus involving Fubini's Theorem Homework Statement f(x,y) = x + y, if: x^2 ≤ y ≤ 2x^2 f(x,y) = 0, otherwise Evaluate \iint_\textrm{I}f where I = [0,1] x [0,1] Homework Equations For a Jordan domain K in ℝ^n, let h: K → ℝ and g: K → ℝ be continuous...

Homework Statement Find the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 that is revolved about the line x=1. The Attempt at a Solution dV=∏R2t where : t=dy R=1-(y-y3) =1-y+y3 so.. dV=∏(1-y+y3)2dy dV=∏(1-2y+y2+2y3-2y4+y6)dy V=∏∫ from 0 to 1 of...

what is the area inside the graph of r=2sinθ and outside the graph of r=sinθ+cosθ? so i compute for the values of 'r',... but, i only got one intersection point which is (45°, 1.41). there must be two intersection points right? but I've only got one. what shall i do? i cannot compute for...

Homework Statement Find the area between y= 1/(x2+1) and the x-axis, from x=0 to x=1 The Attempt at a Solution so when x=0, y=1 and when x=1, y=1/2 next i plot the points, so the intersection of the given equation is (0,1) and (1,1/2) Yh= Y-higher= 1/(x2+1) Yl= Y-lower= 0 the...

Homework Statement 2 ∫ dx / (x+1)√[2x(x+2)] 1 Homework Equations Let x = tan θ if √(a^2 + x^2) Where a = constant The Attempt at a Solution 2 ∫ dx / (x+1)√(2x)√(x+2) 1 2 1/√2 ( ∫ dx / (√x)(x+1)[√(x+2)] 1 Now make all x...

Homework Statement find the following integrals. Homework Equations 1. ∫2x^3 sin(x^2)dx 2.∫ x(2x-3)^1/3dx The Attempt at a Solution Using U-Sub for number 2 i ended with ((3(2x-3)^7/3)/28)+((9(2x-3)^4/3)/16)+C I apologize for the form but I'm new here and don't really know...

And the first thing I'm tackling is understanding the area of a shape by inscribed/circumscribed polygons. For example, the shape produced by the line: y = x + 1, the y axis, and the vertical line x = 2. Now, my understanding is that the area is the limit as n approaches infinity of the...

Homework Statement Find the antiderivative of: \int \frac{dx}{x^3 - 2x^2 + 4x - 18} Homework Equations This was asked in my Calculus II class right after we finished dealing with Solving for Integrals using Partial Fractioning. The Attempt at a Solution This is really more of an algebra...

Homework Statement Hey, just wondering how I might go about doing this problem, as I really have very little idea... Prove the following inequality: \frac{1}{e}\leq\frac{1}{4\pi^{2}}\int_{R}e^{cos(x-y)}dxdy\leqe (hopefully this reads "one over e is less than or equal to one over four pi...

Hi, I have another question that I am having trouble with: Homework Statement Find the value of a such that: \int_{0 }^{a } cos^2 (x) dx The Attempt at a Solution\int_{0}^{a}\frac{1}{2} (1+cos (2x)) \frac{x}{2} + \frac{(sin(2x)}{4} \frac{a}{2} + \frac{sin(2a)}{4} - 0 - 0 = 0.740...

Homework Statement Find the values of a > 0, such that \int_{a }^{a^2} \frac{1}{1+x^2} dx = 0.22 The Attempt at a Solution I integrated it and got: \int_{a }^{a^2} \frac{1}{1+x^2} dx = 0.22 arctan(a^2) - arctan(a) = 0.22 I am stuck after this, how would I solve this :-/ Thanks

Just for fun I'd like to find the equation r(t) which defines the position of an apple as a function of time falling from a distance Ro (radius of the earth) down to a basketball with the mass of the earth--probably a black basketball hole. All websites show the equations for a distance where...

Hello all, I wanted to know whether the following two relations are same. \int x\;f(x)\;{\rm d}x = {\rm A} and \int (x-{\rm A})\;f(x)\;{\rm d}x = 0 are same? 'A' is some positive number, distribution function f(x) is normalized to 1. (i am trying to understand some details of first moment...

Homework Statement Let A be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2pi/n, show that A = (\frac{1}{2})nr^{2}sin(2(\pi)/n) Homework Equations Area of a circle = pi x r^2 Area of...

I'm a Physics/Math major- and am setting up my degree plan I've posted a similar thread before but now I only have one math elective left (and a boatload of choices, all of which sound interesting) I've narrowed it down to either: Integral Calculus, Topology, or Theory of Ordinary...

What's difference between those Courant's books? They both seems to be first year calc books.

Homework Statement \int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx Homework Equations The Attempt at a Solution \int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx Okay I let... u=tanx du=sec^2x Then I got \int \frac{ u }{ (u+4)(u+2) } du Then...

Hi everyone - first a thanks to all who helped me through differential calculus and limits etc - think I'm getting the hang of it... Anyway as you can probably guess the next topic is integration. I'm kinda stuck on the 'concept' of reduction formula. I've done the usual integration...

Consider the tetrahedron which is bounded on three sides by the coordinate planes and on the fourth by the plane x+(y/2)+(z/3)=1 Now the question asks to find the area of the tetrahedron which is neither vertical nor horizontal using integral calculus (a double integral)? I think they mean...

hi there! I'm having some troubles regarding this question: "if a force of 5 pounds produces a stretch of 1/10 of the natural length, L , of the spring, how much work is done in stretching the spring to double its natural length?" i tried answering this but I'm not sure if my answer is...

It has been a while since I have had to do an integration such as this, it is probably quite simple lol. But could someone show me how to integrate (secx)^2, that is secx all squared. I know that the answer is tanx but i was wondering if someone could show me the method. Thank you.

I have taught myself derivatives and I have taught myself the power rule and substitution but the book that I have bin using doesn’t go more in depth than that so what should I learn next?

Hi everyone, this is my first time posting on these forums. If I am doing anything wrong, please let me know. I am having a lot of trouble with conceptualizing rotating lines around the X- and Y-axis. The problem I am trying to visualize right now is... Using integrals to represent...

As a Grade 12 student that is often required to find the roots of quadratics for math, physics, and chemistry problems, I wondered whether there would be any methods for solving these problems excepting the quadratic formula. I was pondering the implications of calculus in algebra and, although...