Integrals Definition and 1000 Threads

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
The integrals enumerated here are those termed definite integrals, which can be interpreted formally as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

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  1. T

    Bounded regions and triple integrals

    Homework Statement a) sketch the region in the first octant bounded by the elliptic cylinder 2x^2+y^2=1 and the plane y+z=1. b) find the volume of this solid by triple integration. Homework EquationsThe Attempt at a Solution I have already sketched the elliptic cylinder and the plane. my...
  2. D

    How Do I Set Bounds for These Integrals Correctly?

    Homework Statement I am having trouble setting up the bounds on the following two integrals: (a) The region E bounded by the paraboloid y=x2+z2 and the plane y=4. (b) The region bounded by the cylinder x2+y2=1, z=4, and the paraboloid z=1-x2-y2. Homework EquationsThe Attempt at a Solution I...
  3. C

    MHB Having Trouble with Boundaries for Triple Iterated Integrals?

    I am having some trouble with finding the boundaries for the first part of the problem (dz dy dx), I should be able to figure out the second part on my own. The problem is: Set up the triple iterated integrals (using dz dy dx and d θ dr dz) to find ∫∫∫E \sqrt{x^2+y^2} dV where E is the part of...
  4. D

    Splitting up an interval of integration

    How does one prove the following relation? \int_{a}^{b}f(x)dx= \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx Initially, I attempted to do this by writing the definite integral as the limit of a Riemann sum, i.e. \int_{a}^{b}f(x)dx=...
  5. I

    MHB Double & Triple Integrals: Same Solution?

    when taking double or triple integrals, do you get the same solution no matter which variable you integrate with respect to first?
  6. B

    Integrals of Complex Functions

    Homework Statement Suppose we have the function ##f : I \rightarrow \mathbb{C}##, where ##I## is some interval of ##\mathbb{R}## the functions can be written as ##f(t) = u_1(t) + i v(t)##. Furthermore, suppose this function is integral over the interval ##a \le t \le b##, which can be found by...
  7. G

    How Do You Determine the Limits for Surface Integrals?

    Homework Statement Homework Equations ∫∫D F((r(u,v))⋅(ru x rv) dA The Attempt at a Solution [/B] I got stuck after finding the above, at where the double integrals are. :( May I know how do I find the limits of this? (I always have trouble finding the limits to sub into the integrals...
  8. L

    Questions concerning integrals in Schwartz's QFT text

    Two (supposedly) trival questions in Schwartz's QFT notes. The notes can be found http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf. 1. page 155, equation 15.2, how does the integrand reduce to k dk? I would guess that there must be some logarithm, but k dk? 2. page 172...
  9. C

    The Fundamental Theorem for Line Integrals

    Homework Statement Determine whether or not f(x,y) is a conservative vector field. f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) > If F is a conservative fector field find F = gradient of f Homework Equations N/A The Attempt at a Solution Fx = -3e^(-3x)(-3)cos(-3y) Fy =...
  10. J

    MHB Evaluating Definite Integrals with Floor Function

    Evaluation of \displaystyle \int_{0}^{\pi}\lfloor \cot x \rfloor dx and \displaystyle \int_{0}^{\pi}\lfloor \cos x \rfloor dx\;, where \lfloor x \rfloor denote Floor function of x
  11. L

    Need some help with some various integrals (studying for finals)

    < Mentor Note -- thread moved from General Math to the Homework Help forums >Hi all, Calc II finals is 4-5 weeks away...We're on Taylor Series right now, but I wanted to get started early on studying for the final. I have a few questions that are confusing me that I took from a final exam I saw...
  12. G

    Solving the Homework Puzzle: Finding Your Error with Surface Integrals

    Homework Statement The problem is given in the attached file. Homework Equations Divergence theorem, flux / surface integral The Attempt at a Solution [/B] As you can see I got the question correct using Divergence theorem. But I wanted to make sure that I could arrive at the same answer...
  13. A

    How to use contour, complex analysis to solve integrals?

    Homework Statement \int_{-\infty}^{\infty} \frac{\sin(x)}{x} using Complex Analysis Homework Equations Contour analysis on \int_{-\infty}^{\infty} \frac{\sin(x)}{x} The Attempt at a Solution Hello, I am completely new to contour integration. I would really appreciate it if someone can walk...
  14. G

    Surface Integrals of first octant

    Homework Statement Evaluate ∫∫ F⋅dS, where F = yi+x2j+z2k and S is the portion of the plane 3x+2y+z = 6 in the first octant. The orientation of S is given by the upward normal vector. Homework Equations ∫∫S F⋅dS = ∫∫D F(r(u,v))⋅||ru x rv|| dA, dA=dudv The Attempt at a Solution [/B] Since...
  15. _N3WTON_

    Double Integrals, Jacobians, Thermodynamics

    Homework Statement An important problem in thermodynamics is to find the work done by an Ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region R enclosed by two isothermal curves xy = a...
  16. A

    Solving Integrals using summations

    Homework Statement Many places I have seen when solving integrals you change a lot of it into sums. http://math.stackexchange.com/questions/1005976/finding-int-0-pi-2-dfrac-tan-x1m2-tan2x-mathrmdx/1006076#1006076 Is just an example. So in general, how do you solve integrals (CLOSED FORM) by...
  17. sheldonrocks97

    Find constants that satisfy integrals?

    Homework Statement ∫y1(x)^2dx from - to + infinity=1 and ∫y2(x)^2dx from - to + infinity=1 Homework Equations None that I know of. The Attempt at a Solution I evaluated the integrals and got that c1 is equal to c2 but I think that's wrong.
  18. A

    MHB Interchanging Summation and Integrals?

    Hello, Suppose we have: $$\begin{align} \sum_{n=1}^{\infty}\frac{1}{9n^2 + 3n - 2} &=\frac{1}{3}\sum_{n=1}^{\infty}\left(\frac{1}{3n - 1}-\frac{1}{3n + 2}\right)\\\\ &=\frac{1}{3}\sum_{n=1}^{\infty}\int_0^1\left(x^{3n-2}-x^{3n+1}\right){\rm d}x\\\\...
  19. PhysicsKid0123

    Confused about force and work in 3 Dimensions. Line integrals.

    So I am kind of confused about the role of force when calculating work. Specifically, when defining work using a line integral. There is a paragraph in my calculus book that is really throwing me off and its really bugging me so much I can't continue reading unless I fully understand what's...
  20. A

    MHB Evaluating infinite sum for e^(-x) using integrals

    Hello, I have began my journey on infinite sums, which are very interesting. Here is the issue: I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle...
  21. D

    Physics interpretation of integrals of differential forms

    Be a vector field \vec{F}=(f_1,f_2,f_3) and \omega^k_{\vec{F}} the k-form associated with it , i know if i do \int \omega^1_{\vec{F}} is the same of a line integral and \int \omega^2_{\vec{F}} i obtain the same result of \int \int_S \vec{F}\cdot d\vec{S}, which is the flux of a vector field in a...
  22. S

    MHB Integrals at infinity/ factorials problem

    Need help on exercise 2 from the linked image , left first in so you guys could see the Γ(χ) function any help is appreciated , thanks in advance!
  23. DavideGenoa

    Comparing 2 Improper Integrals: Convergence & Criteria

    I read that the improper Riemann integral ##\int_0^1 \frac{1}{x}\sin\frac{1}{x}dx## converges and that ##\int_0^1 |\frac{1}{x}\sin\frac{1}{x}|dx## does not. I have tried comparison criteria for ##\int_0^1 |\frac{1}{x}\sin\frac{1}{x}|dx##, but I cannot find a function ##f## with a divergent...
  24. _N3WTON_

    Proving Properties of Double Integrals

    Homework Statement Prove the following property: If m <= f(x,y) <= M \hspace{2 mm} \forall (x,y) \in D, then: mA(D) <= \int\int f(x,y)\,dA <= MA(D) Homework Equations I use a few other known properties in the proof (see below) The Attempt at a Solution First, I should state that this problem...
  25. _N3WTON_

    How can I evaluate this double integral?

    Homework Statement Evaluate the following double integral: V = \int\int \frac{3y}{6x^{5}+1} \,dA D = [(x,y) \hspace{1 mm}|\hspace{1 mm} 0<=x<=1 \hspace{5 mm} 0<=y<=x^2] Homework EquationsThe Attempt at a Solution V = \int_{0}^{1} \int_{0}^{x^2} \frac{3y}{6x^{5}+1}\,dy\,dx =...
  26. P

    Can You Solve These Tricky 2D Integrals on a Unit Circle?

    I can't compute the integral: \int \frac{\arccos(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}\frac{x-a}/{(\sqrt{(x-1)^2+y^2})^3 dxdy on an unit circle: r < 1. for const: a = 0.01, 0.02, ect. up to 1 or 2. I used a polar coordinates, but the values jump dramatically in some places (around the 'a' values)...
  27. J

    Integrals of (-3csc(theta))/(1+cos(theta))

    Homework Statement integrate (-3csc(theta))/(1+cos(theta)) Homework Equations i'm not sure The Attempt at a Solution i tried using u sub. but i got nowhere. U=1+costheta Du=-sintheta
  28. H

    Definite Integrals Using Contour Integration

    Problem Show: \int_0^\infty \frac{cos(mx)}{4x^4+5x^2+1} dx= \frac{\pi}{6}(2e^{(-m/2)}-e^{-m}) for m>0 The attempt at a solution The general idea seems to be to replace cos(mx) with ##e^{imz}## and then use contour integration and residue theory to solve the integral. Let ##f(z) =...
  29. L

    Calculating Arc Length of a Curve: A Calculus II Problem

    Homework Statement Find the exact length of the curve: y= 1/4 x2-1/2 ln(x) where 1<=x<=2 Homework Equations Using the Length formula (Leibniz) given in my book, L=Int[a,b] sqrt(1+(dy/dx)2) I found derivative of f to be (x2-1)/2x does that look correct? The Attempt at a Solution I found f'...
  30. L

    Definite integrals with trig issues

    from 0 to π/2 ∫sin5θ cos5θ dθ I have been trying to solve the above for quite some time now yet can't see what I am doing wrong. I break it down using double angle formulas into: ∫ 1/25 sin5(2θ) dθ 1/32 ∫sin4(2θ) * sin(2θ) dθ 1/32 ∫(1-cos2(2θ))2 * sin(2θ) dθ With this I can make u = cos(2θ)...
  31. S

    How can I solve for average velocities using derivatives and integrals?

    Stressed first year university student here, fresh out of high school. I took physics in both grade 11 and 12, and thought I had a pretty good grasp on it; that is until this week. Introduction to derivatives and integrals to get from x(t) to v(t) to a(t) and vice-versa. I have a pretty good...
  32. DivergentSpectrum

    Question about nonelementary integrals

    are nonelementary integrals implicit functions? ie, when we do implicit differentation, we get an explicit function. What if i go the opposite way, and integrate an explicit function to get an implicit antiderivative?
  33. G

    Finding Integrals of Non-Elementary Functions

    Homework Statement We know that F(x) = \int^{x}_{0}e^{e^{t}} dt is a continuous function by FTC1, though it is not an elementary function. The Functions \int\frac{e^{x}}{x}dx and \int\frac{1}{lnx}dx are not elementary funtions either but they can be expressed in terms of F. a)...
  34. RJLiberator

    Does Integral Zero Imply Function Zero?

    True or False? Let a and b be real numbers, with a < b, and f a continuous function on the interval [a, b]. a) If a=b then \int^{b}_{a} f(x)dx = 0 My answer: This is TRUE, because while this integral would have a height, it would NOT have a width and area being l*w will result in 0. b) If a...
  35. L

    Evaluating definite integrals for the area of the regoin

    Homework Statement Evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. f(x) = \frac{1}{x^2+1} at the point (1,1/2) Homework Equations The Attempt at a Solution So far I...
  36. Dethrone

    MHB How can we recognize standard integrals here

    Also, how would you do $\int \sqrt{x^2-4}$? $$d(x^2-4)^{3/2}=3x\sqrt{x^2-4} \,dx$$ $$\int \frac{\sqrt{x^2-4}}{3x}d(x^2-4)^{3/2}$$ Not sure how partial integration will be useful here. What standard integrals do you see?
  37. G

    Can you help me prove the integral for Hermite polynomials?

    Hi. I'm off to solve this integral and I'm not seeing how \int dx Hm(x)Hm(x)e^{-2x^2} Where Hm(x) is the hermite polynomial of m-th order. I know the hermite polynomials are a orthogonal set under the distribution exp(-x^2) but this is not the case here. Using Hm(x)=(-1)^m...
  38. Mr-R

    Can the Tensor Integral in GR be Bounded by a Region Outside of d^{3}x?

    Dear all, I am self studying GR and stuck on problem (23) on page 108/109. I am trying to do all of them. First I will start with (a) so you guys can breath while laughing at my attempts at (b) and (c) :blushing: (a) Attempt The tensor in the equation is bounded in the d^{3}x region. Outside...
  39. F

    When Integrating (2x)/(4x^(2)+2) I get two different integrals ?

    Hi So let's have ∫(2x)/(4x^(2)+2) dx Without factorising the 2 from the denominator, I integrate and I get 1/4*ln(4x^(2)+2)+c which makes sense as when I differentiate it I get the original derivative. BUT when I factor the 2 from the denominator I have 2x/[2(2x^(2)+1)]...
  40. B

    Why do some integrals require correction for extreme precision?

    I have read that even the simplest integrals (like y=x2) might need some correction if we want to reach an extreme precision. Is that really so? Can you explain why or give me some useful links? Thanks
  41. K

    Path Integrals in QFT: Beyond Peskin's Reference

    Can anyone suggest me a good reference for path integrals (QFT), apart from peskin.
  42. davidbenari

    Understanding the Rationale Behind Flux Integrals

    Ok for the purpose of this question let's stick to the flux integral: The general formula is ∫∫s (E-vector)*(dS-vector)=Flux where * stands for the dot-product. Now, I like it when my integrals make sense, and to do that I usually think of the Riemann Sum which might represent my integral...
  43. M

    Hermite representation for integrals?

    Suppose I want an expectation value of a harmonic oscillator wavefunction, then in what way will I write the Hermite polynomial of nth degree into the integral? I have a link of the representation, but don't know what to do with them? http://dlmf.nist.gov/18.3
  44. J

    Line integrals, gradient fields

    Homework Statement ##\nabla{F} = <2xyze^{x^2},ze^{x^2},ye^{x^2}## if f(0,0,0) = 5 find f(1,1,2)Homework Equations The Attempt at a Solution my book doesn't have a good example of a problem like this, am I looking for a potential? ##<\frac{\partial}{\partial x},\frac{\partial}{\partial...
  45. Dethrone

    MHB Why Split Improper Integrals?

    From textbooks, I usually see that when there is an integral like this: \int_{-\infty}^{+\infty} f(x)\,dx, they generally split it two, usually by 0. \int_{-\infty}^{0} f(x)\,dx + \int_{0}^{\infty} f(x) \,dx They do the same for points of discontinuity, but if you notice, the number that they...
  46. J

    Mastering Double Integrals: Solving Problems with Ease

    Homework Statement Set up the double integral over the region ##y=x+3; y=x^2+1## Homework Equations The Attempt at a Solution finding the intersections you get the double integral ##\int_{1}^{5}\int_{-1}^{2}dxdy =12 ## but why is that not the same as...
  47. J

    Iterated integrals over region w

    Homework Statement I am given W = \{ (x,z,z)| \frac{1}{2} \le z \le 1; x^2 + y^2 +z^2 \le 1\} they want the iterated integrals to be of the form \iiint_W dzdydx The Attempt at a Solution so I know z=1/2 will give me the larger bound for x x^2 + y^2 + (1/2)^2 =1...
  48. Greg Bernhardt

    What are the Standard Integrals?

    Definition/Summary This article is a list of standard integrals, i.e. the integrals which are commonly used while evaluating problems and as such, are taken for granted. This is a reference article, and can be used to look up the various integrals which might help while solving problems...
  49. J

    Double integrals interchanging order

    Homework Statement \int_{1}^{4}\int_{1}^{\sqrt{x}}(x^2+y^2)dydx The Attempt at a Solution I drew the region, I tried \int_{1}^{2}\int_{1}^{y^2}(x^2+y^2)dxdy but it doesn't seem to work. when the order is changed 1 \le y \le 2 and \sqrt{x} = y \rightarrow...
  50. Mogarrr

    Improper Integrals of Odd functions

    I think I may have found an error in the text I'm reading. Here's a quote: ... + \int_0^{\infty}x^rf_1(x)sin(2\pi logx)dx. However, the transformation y=-logx-r shows that this last integral is that of an odd function over (-∞,∞) and hence equal to 0 for r=0,1,... By the way, the author means...
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