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. A

    MHB Basic Calculus II Integral Questions - Riemann Sums, Absolute Integrals, etc.

    Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...
  2. S

    Stokes's Theorem showing 2 surface integrals are equal

    Homework Statement Let F = <z,x,y>. The plane D1: z = 2x +2y-1 and the paraboloid D2: z = x^2 + y^2 intersect in a closed curve. Stoke's Theorem implies that the surface integrals of the of either surface is equal since they share a boundary (provided that the orientations match)...
  3. erzagildartz

    How to Integrate Triple Integrals in Different Coordinate Systems?

    how to solve triple integrals in cylindrical, spherical and rectangular coordinates ..easy ways
  4. B

    Change of variables in double integrals

    I know the formula for a change of variables in a double integral using Jacobians. $$ \iint_{S}\,dx\,dy = \iint_{S'}\left\lvert J(u,v) \right\rvert\,du\,dv $$ where ## S' ## is the preimage of ## S ## under the mapping $$ x = f(u,v),~ y = g(u,v) $$ and ## J(u,v) ## is the Jacobian of the mapping...
  5. J

    Website title: Formal Relation Between Integrals of y=f(x) and y=x

    Is there a formal relation that links \int yxdx OR \int_{a}^{b}yxdx with \int xydy OR \int_{a}^{b}xydy where y=f(x) over the interval x\in\left[a,b\right].
  6. D

    Subdividing an integral into a sum of integrals over a given interval

    How does one prove the following: \int^{c}_{a} f\left(x\right)dx = \int^{b}_{a} f\left(x\right)dx +\int^{c}_{b} f\left(x\right)dx where f\left(x\right) is continuous in the interval x\in \left[a, b\right], and differentiable on x\in \left(a, b\right). My approach was the following...
  7. R

    What is the simplified integral for evaluating double integrals with IBP?

    Homework Statement ∫∫[ye^(-xy)]dA R=[0,2]×[0,3] evaluate the integral. Homework Equations The Attempt at a Solution So I started with some algebra changing the integral to ∫(e^-x)[∫ye^-ydy]dx I evaluated the y portion first because its more difficult to deal with and wanted to...
  8. S

    MHB Norm of Integrals: Bounding the Matrix Product

    Hi I have an integral over [0,1] of product of two matrices say A(t). B(t) and I wish to bound its norm. Can you say that ||integral (AB)||<||B(t)||.||integral (A)|. is there some conditions on that to occur thanks sarrah
  9. E

    IBP Struggles: Solving Integrals of \frac{x^2}{e^x+1} \& \frac{x^3}{e^x+1}

    Homework Statement Find the Integrals of \frac{x^2}{e^x+1}\\ \frac{x^3}{e^x+1} Homework Equations Integration by parts The Attempt at a Solution I did IBP twice and it seemed to just get bigger and uglier and now I am stuck. I found the solutions online of the integrals but...
  10. E

    Equality of definite integrals, relation between integrands

    Suppose we are given two functions: f:\mathbb R \times \mathbb C \rightarrow\mathbb C g:\mathbb R \times \mathbb C \rightarrow\mathbb C and the equation relating the Stieltjes Integrals \int_a^\infty f(x,z)d\sigma(x)=\int_a^\infty g(x,z)d\rho(x) where a is some real number, the...
  11. L

    Infinite series and improper integrals

    Hello, I've been reviewing some calculus material lately and I just have a couple questions: 1) I've seen infinite series shown graphically as a collection of rectangular elements under a curve representing an approximation of the area under the curve. But the outputs of the infinite...
  12. M

    Maximizing Contour Integrals: Tips and Tricks

    Hello. Question is attached. Can someone check my answer? Did I convert the last part correctly? Is everything correct?
  13. F

    MHB Complex Integrals (for me at least)

    Hey! How do I integrate ∫tln√(t+1) and ∫4te^(2-0,3t)? Thanks in advance.
  14. DreamWeaver

    MHB Derivatives and Integrals of the Hurwitz Zeta function

    Initially, the purpose of this tutorial will be to explore and evaluate various lower order derivatives of the Hurwitz Zeta function. In each case, the Hurwitz Zeta function will be differentiated with respect to its first parameter. A little later on - although this will take some time! - these...
  15. Saitama

    MHB Comparing fractions with definite integrals

    Hello! I found the following problem on AOPS: Which is larger, $$\Large \frac{\int_{0}^{\frac{\pi}{2}}x^{2014}\sin^{2014}x\ dx}{\int_{0}^{\frac{\pi}{2}}x^{2013}\sin^{2013}x\ dx}\ \text{or}\ \frac{\int_{0}^{\frac{\pi}{2}}x^{2011}\sin^{2011}x\ dx}{\int_{0}^{\frac{\pi}{2}}x^{2012}\sin^{2012}x\...
  16. B

    Surface integrals to derive area of sphere

    Given a sphere x^2 + y^2 + z^2 = a^2 how would I derive the surface area by using surface integrals? The method I've tried is as follows: dA = sec\ \gamma \ dxdy where gamma is the angle between the tangent plane at dA and the xy plane. sec \gamma = \frac{|\nabla \varphi|}{\partial \varphi...
  17. B

    MHB Triple Integrals in Spherical Coordinates

    Hi all, I'm not sure how to get the boundaries in terms of both the spherical and cylindrical coordinates for this question. Here are the boundaries we were given in the solution. How was \frac{\pi}{4} for φ and \frac{1}{\sqrt{2}} for r obtained? Thanks!
  18. Thor Shen

    About two integrals in QCD textbook by muta

    http://d.kankan3d.com/file/data/bcs/2014/0508/w65h1446064_1399517186_873.jpg 1.How to deal with the delta functions in eq.2.3.153 to obtain the eq.2.3.154 by integrating over q'? 2.How to caculate the integral from eq.2.3.154 to eq.2.3.156, especially the theta function?
  19. J

    Trigonometric integrals; choosing which one to break up?

    trigonometric integrals; choosing which one to "break up?" When you have two different trigonometric functions multiplied together within the integral, for example integral of (cos^4*sin^6) how do you tell which one to "break them up" to substitute an identity in? Thank you!
  20. T

    Double Integrals using Polar Coordinates

    Homework Statement ∫∫Rarctan(y/x) dA, where R={(x,y) | 1\leqx2+y2\leq4, 0\leqy\leqx Homework Equations x=rcos(θ) y=rsin(θ) x2+y2=r2 The Attempt at a Solution I know that the range of r is 1 to 2 but I can't figure out how to change the second part into θ. If I change y and x to...
  21. ShayanJ

    Integrals containing (x^2+a^2-2xa cos(theta))^(-1/2)

    Integrals containing \frac{1}{\sqrt{x^2+a^2-2xa \cos{\theta}}} occur frequently in physics but I still have problem solving them. Is there a general method for dealing with them?(Either w.r.t. x or \theta ) Thanks
  22. L

    Quick question on double/triple integrals for area and volume

    I do not know how to formulate formulas on this forum so I just wrote it neatly on a piece of paper and linked it. http://puu.sh/8fwXr.jpg Thankss.
  23. Digitalism

    Calculators Definite Double Integrals 2 variables TI-89 Titanium

    Hello I am trying to solve this integral 25-9x^2-25y^2/9 dydx integrating from 0 to sqrt(9-9x^2/25) and the limits of the second integration are 0 to 5. I can find tutorials on how to find the definite double integral of a single variable, but not for two variables. Any clues? edit: so far...
  24. Rectifier

    Integrals giving me a hard time

    Hey! It is the first time I post on this subform. Please forgive me if I do something wrong. Homework Statement F(x)=\int^x_0f(t)dt for R \in t \rightarrow f(t) Homework Equations Is it true that 0 \leq f(x)\leq 3 for 0<x<1 \int^x_0 tf(t)dt \leq x^2 for all x \in (0, 1)? "The Attempt at...
  25. J

    Are These Formulas for Indefinite Integrals and Antiderivatives Correct?

    \int \frac{d}{dx}f(x)dx = f(x) + C_x \iint \frac{d^2}{dx^2}f(x)dx^2 = f(x) + xC_x + C_{xx} \int \frac{\partial}{\partial x}f(x,y)dx = f(x,y) + g_x(y) \int \frac{\partial}{\partial y}f(x,y)dy = f(x,y) + g_y(x) \iint \frac{\partial^2}{\partial x^2}f(x,y)dx^2 = f(x,y) + x g_{x}(y) + g_{xx}(y)...
  26. R

    MHB Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values

    I think you may be interested in the Wolfram Demonstration, "Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values"
  27. J

    Finding an integral given two other integrals?

    Homework Statement It is given that integral(1 to 2) g(x)dx=22 integral (1 to 4) g(x)dx=7 integral (1 to 16) g(x)dx=13 Find integral (4 to 16) Homework Equations Using properties of integrals, integral(4 to 16)= integral(1 to 16) - integral(1 to 4) The Attempt at a...
  28. S

    Real integrals using complex analysis

    Homework Statement After successfully solving a lot of integrals I gathered 4 ugly ones that I can not solve: a) ## \int _{-\infty} ^\infty \frac{cos(2x)}{x^4+1}dx## b) ##\int _0 ^\infty \frac{dx}{1+x^3}## c) ##\int _0 ^\infty \frac{x^2+1}{x^4+1}dx## d) ##\int _0 ^{2\pi } \frac{d\varphi...
  29. Feodalherren

    Surface integrals - parametrizing a part of a sphere

    Homework Statement Find the area of the part of the sphere x^2 + y^2 + z^2 = 4z that lies inside the paraboloid x^2 + y^2 = z Homework Equations The Attempt at a Solution I solved for the intercepts and found that they are z=0 and z=3. The sphere is centered two units in the z-direction above...
  30. S

    Solving Complex Integrals with Cauchy's & Residue Theorem

    Homework Statement Calculate following integrals: a) ##\int _{|z|=1}\frac{e^z}{z^3}dz## b) ##\int _{|z|=1}\frac{sin^6(z)dz}{(z-\pi /6)^3}## Homework Equations The Attempt at a Solution I am really confused, so before writing my solutions I would need somebody to please tell me: - What is...
  31. pellman

    Path integrals as usually presented - what does it tell us?

    Typical introductions to path integrals start with asking for the value of \langle x_1,t_1 | x_2,t_2 \rangle. This is usually interpreted as the probability amplitude of observing a particle at x_2 at time time t_2 given that it is located at x_1 at t_1. But is this so? I am having trouble...
  32. P

    Problem with unknown functions in integrals

    Homework Statement I often have a problem dealing with unknown functions in derivations. Recently I was looking at variance of pdf's and tried to do the integral below with no success. Could someone suggest a method, or point out where I am going wrong. Homework Equations Show ∫(x - μ)2...
  33. S

    Calculate real integrals using complex analysis

    Homework Statement Calculate real integrals using complex analysis a) ##\int_{-\infty}^{\infty}\frac{dx}{x^2+1}## b) ##\int_0^\infty \frac{sin(x)}{x}dx##Homework Equations The Attempt at a Solution a) ##\int_{-\infty }^{\infty }\frac{dz}{z^2+1}=\int_{-R}^{R}\frac{dx}{x^2+1}+\int...
  34. M

    Finding Integrals: ∫ (5x^2 + sqrt(x) - 4/x^2) dx

    I have these integrals to find: ∫ (5x^2 + sqrt(x) - 4/x^2) dx ∫ [cos(x/2) - sin(3x/2)] dx ∫ s/sqrt(s^2 + 4) ds (upper coordinate is 5 lower coordinate is 1) I have worked it out as: ∫〖(5x^2+√x〗-4/x^2) dx=5x^3/(2+1)+x^(1/2+1)/(1+1/2)-4x^(-2+1)/(-2+1)+C=5/3 x^3+2/3x^(3/2)+4/x+C...
  35. M

    When using stokes theorem to remove integrals

    hey pf! i had a question. namely, in the continuity equation we see that \frac{\partial}{\partial t}\iiint_V \rho dV = -\iint_{S} \rho \vec{v} \cdot d\vec{S} and we may use the divergence theorem to have: \frac{\partial}{\partial t}\iiint_V \rho dV = -\iiint_{V} \nabla \cdot \big( \rho...
  36. J

    The true TFC for surface integrals

    The true FTC for surface integrals Let's say that ##\vec{f}## is an exact one-form, so we have that ##\vec{f}=\vec{\nabla}f##, and ##\vec{F}## is an exact two-form, so we have that ##\vec{F}=\vec{\nabla}\times \vec{f}##. The fundamental theorem of calculus for line integral says that...
  37. E

    Help with Trigonometric Integrals

    Could someone help me with these two problems? I've been at them for an hour, but have very little clue how to go about solving either of them. Homework Statement 1)∫ 6 csc^3 (x) cot x dx Homework Equations The Attempt at a Solution 6 ∫ csc^3 (x) dx) / tan x csc^3 / tan x =...
  38. J

    Calculus Definite Integrals: Volumes by Washer Method

    Homework Statement Using Washer Method: Revolve region R bounded by y=x^2 and y=x^.5 about y=-3 Homework Equations V= integral of A(x) from a to b with respect to a variable "x" A(x)=pi*radius^2 The Attempt at a Solution pi(integral of (x^.5-3)^2 -(x^2)^2-3) from 0 to 1 with...
  39. Y

    Fourier Integrals and Division

    Homework Statement (a) Find the Fourier transform f(ω) of: f(x) = cos(x) between -pi/2 and pi/2 (b) Find the Fourier transform g(ω) of: g(x) = sin(x) between = -pi/2 and pi/2 (c) Without doing any integration, determine f(ω)/g(ω) and explain why it is so Homework Equations f(ω) =...
  40. AntSC

    Indefinite integrals with different solutions?

    Indefinite integrals with different solutions? Homework Statement \int \csc ^{2}2x\cot 2x\: dx Solve without substitution using pattern recognition Homework Equations As above The Attempt at a Solution To try a function that, when differentiated, is of the same form as the...
  41. J

    Is f(x) an antiderivative of f'(x) or a family of antiderivatives?

    By FTC, every function f(x) can be expessed like: f(x) = \int_{x_0}^{x}f'(u)du + f(x_0) Now, I ask: f(x) is a antiderivative of f'(x) or is a family of antiderivative of f'(x) ?
  42. J

    Understanding how to set up integrals for inertia

    Hello, and thank you in advanced for this. I am having trouble with setting up most if not all of my integrals when I am trying to find the elements of an inertia tensor. What would I do if i need to find say the tensor for a disk, but i don't know what to take for my three limits to be. i get...
  43. P

    Improper Integrals: Solve ∫-∞ to ∞ e^-|x| dx

    Homework Statement \int_{-\infty}^{+\infty} e^{-\left|x\right|} \,dx The Attempt at a Solution So I know you are supposed to split this integral up into two different ones, from (b to 0) and (0 to a) where b is approaching - infinity, and a is approaching +infinity, but how would I take...
  44. Jewish_Vulcan

    Learning integrals and derivitaves in pre calculus.

    Hello I am in pre-calculus which is the next math class after algebra 2 and there are many scientific equations that require a knowledge of calculus to solve. For example I do science olympiad maglev and many of the equations to solve for magnetic flux or magnetic fields etc.. use derivatives...
  45. KingCrimson

    Why do we write dx in indefinite integrals

    in understand why we write the dx in riemann integral , but in the indefinite integral why do we use that ? what is the relation between the area under a curve , and the antiderivative of that of that curve ??
  46. S

    Substituting differentials in physics integrals.

    Today I tried to show that rotational kinetic energy was equivalent to standard translational kinetic energy. So I started with kinetic energy, T = ∫dT. Then, because T=1/2mv^2, I substituted dT=1/2v^2dm and then because m=ρV, I substituted dm=ρdV. Then, after substituting v=ωr, I got the...
  47. T

    Solving for the Volume of a Solid Using Double Integrals

    Homework Statement Find the volume of the solid bounded above by the surface z = x^2 + y^2 and below by the triangular region in the xy-plane enclosed by the lines x = 0 , y = x , and x + y = 8. Homework Equations V = ∫∫ Height Base The Attempt at a Solution I first found...
  48. matqkks

    Improper Integrals: Real-Life Applications & Syllabus Impact

    What are the real life applications of improper integrals? Why are they on the syllabus of every first course in calculus? I am looking for examples which have a real impact.
  49. J

    Are residues useful for proper integrals?

    Calculating residues are useful when we are trying to solve some improper integral, because the Cauchy principal value will be the sum of residues inside the path taken (if the integral along the complex path tends towards 0). When we have a proper integral of trigonometric functions, this is...
  50. X

    Integrals of Complex Functions

    Homework Statement What is the integral from negative infinity to positive infinity of the following functions? a) f(z) = \frac{e^{-i5z}}{z^{2}+1} b) f(z) = \frac{e^{-i5z}}{z^{2}-1} c) f(z) = \frac{1}{π}\frac{a}{z^{2}+a^{2}} d) f(z) = e^{\frac{-(z-ia)^{2}}{2}} e) f(z) = \frac{sinz}{z}...
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