What is Integrals: Definition and 1000 Discussions

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.

View More On Wikipedia.org
Recent contents Top users
  1. G

    How Do Contour Integrals Apply to Green's Functions in Acoustic Wave Equations?

    A question about an integral encountered in a paper I am reading about Green's Functions of the acoustic wave equation ... The integral encountered: Im{Integrate[ exp((i*y-a)*k), dk, 0, Infinity]} = Re{1/(y+ i*a)} where i = sqrt(-1) and a,y,k elements of R. Been a while since I've calculated...
  2. dwn

    Line Integrals and Finding Parametric Equations

    I am having a difficult time finding the parametric equations x = x(t) and y = y(t) for line integrals. I know how to find them when dealing with circles, but when it comes to finding them for anything else, I don't see the method...it all seems very random. I did fine with finding the...
  3. binbagsss

    Sin/cos integrals multiplying results (fourier transform).

    Okay, I am trying to determine the Fourier transform of cos (2\pix)=f(x) Where F(k)=^{\infty}_{\infty}\intf(x)exp^{-ikx} dx, So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as...
  4. Y

    MHB Integrating Complex Integrals: Exploring the Mystery of a "-1

    Hello, I have this integral here: \[\int e^{\sqrt{x}}dx\] and I wanted to ask, why can't I treat it like I would treat this integral: \[\int (3x+5)^{5}dx\] In which I would integrate as if g(x)=3x+5 is a normal x, and then divide by the inner derivative ? I tried it with the upper integral...
  5. A

    Integrals over vector fields and Ampere's Law

    Homework Statement Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle in the plane perpendicular to the wire and whose center is the axis of the wire. Ampere's Law relates the electric current to its magnetic effects and states...
  6. C

    Solving Improper Integrals: 1/sqrt(9-x^2) 0 to 3

    Homework Statement integral of 1/sqrt(9-x^2) from 0 to 3 Homework Equations /// The Attempt at a Solution I integrate it correct to arcsin(x/3) from 0 to 3 Get the correct anwser of pi/2. But there is another question, At which value of x in the integration region [0,3]...
  7. A

    Solving Tricky Integrals: What Technique to Use?

    Homework Statement Which technique i should use to solve these integrals? Homework Equations The Attempt at a Solution
  8. S

    MHB Polynomial Long Division w Integrals

    I do not understand how I would do this with long division since there is only 2 terms. I can't remember the trick. Here is what I have so far. \int \frac{3x^2 - 2}{x^2 - 2x - 8} dx so I got \int 3 + \frac{x^2 - 2}{(x - 4)(x + 2)} I'm not sure if that's right? I just factored it out instead...
  9. R

    Why is the coefficient -2 instead of -2/3 in the improper integral solution?

    Homework Statement \int (x-2)-3/2dx Homework Equations \intf(x)dx from 0 to ∞ = lim (t\rightarrow∞) \intf(x)dx from 0 to tThe Attempt at a Solution I have the solution from the solution manual, but I'm just not sure on one of the steps, after you substitute u=(x-2) and du=dx, then integrate...
  10. W

    Why doesn't my method for integrating sinxcosx work?

    I know the way to do \int sinxcos is by u-substitution but why doesn't the following work? sin(2x) = 2sinxcosx \\ \frac{sin(2x)}{2}=sinxcosx \\ \int sinxcosx= \frac{1}{2} \int sin(2x) = -\frac{cos(2x)}{4}
  11. T

    Trig substitution into integrals

    I was testing for convergence of a series: ∑\frac{1}{n^2 -1} from n=3 to infinity I used the integral test, substituting n as 2sin(u) so here's the question: when using the trig substitution, I realized the upperbound, infinity, would fit inside the sine. Is it still possible to make...
  12. V

    Fresnel Integrals, Contour Integration

    Homework Statement Please let me know if this kind of posting of exact problems from a textbook isn't allowed; if that's the case I'll delete it immediately. From Boas's Mathematical Methods in the Physical Sciences, Third Edition: The Fresnel integrals, \int_0^u sin (u^2)\,du and...
  13. T

    MHB  Solving an Integral Question: Where Did I Mess Up?

    I am working on this question: ∫ [(3+ lnx)^2 (2-ln x)] / (4x) dx My answer is: 18/4 [ 3 (ln x^2/2) + 4 (ln x^3/3) - ln x^4/4 ] + C But the answer from the solutions is (5/12) (3+ ln x)^3 - (1/16)(3+lnx)^4 + C Where did I mess up? ∫ [(3+ lnx)^2 (2-ln x)] / (4x) dx let u =ln x du/dx =...
  14. S

    How Do You Calculate Surface Integrals in Fortran 90?

    I am dumfounded on how one would perform surface integrals in Fortran 90 over a platelet, or a rectangular box. I can do single and double integrals but I have no idea on how to do surface integrals Thanks in advance!
  15. 1

    Getting two different integrals for same function(?)

    Homework Statement The actual problem is ∫sin2x/((sinx)4+(cosx)4) dx Homework Equations The Attempt at a Solution First wrote the expression as ∫\frac{2sin2x}{((sinx)^2+(cosx)^2)^2+((sinx)^2-(cosx)^2))^2 } dx then I changed the 2dx to d(2x)...
  16. S

    MHB Is My Integral Evaluation Correct?

    Evaluate the Integral. Just need someone to check my work. \int sec^4y \, tan^4y \tan^4y * sec^2y * sec^2y \, dx tan^4y * (1 + tan^2y) * sec^2y u = tany du = sec^2y \int u^4 * (1 + u^2) * du \int u^4 + u^6 * du \frac{u^5}{5} + \frac{u^7}{7} + C \frac{tan^5x}{5} + \frac{tan^7x}{7} + C
  17. S

    MHB What is the solution to the definite integral $\int^1_0 x^2 e^x \, dx$?

    Evaluate the following integrals. a) $\int^1_0 x e^x dx$ So integrating by parts we get $u = x $ $vu = e^x dx$ $du = dx$ $ v = e^x$ $uv - \int vdu = x e^x - \int^1_0 e^x dx$ xe^x - e^x |^1_0 = 1 b) \int^1_0 x^2 e^x \, dx Integrating by parts we get u = x^2 dv = e^x dx du = 2xdx...
  18. L

    Question about the symmetry of integrals

    O.K. , this question is inspired by a physics class I'm taking where we're working out the expectation values of wave functions, but I think the question really belongs in the math section. Thank you in advance for any help. Here goes nothing... We have a function ψ(x,y,z) = x e\sqrt{}x2 +...
  19. C

    Antiderivative Definite Integrals

    Homework Statement So I did an entire antiderivative, and ended with this part: sec(x)tan(x) + ln|sec(x) + tan(x)| + C I have to do this with the lower bound of -pi/3 and 0. When I do it, I should be getting 2√3 + ln(2+√3) But, I'm getting (0+0)-(2*-√3 + ln(2-√3)) Which would...
  20. J

    MHB Understanding Work Integrals: Examples Explained

    I have a question about work integrals. I'm trying to reconcile using integrals to essentially multiply force by distance, but the fact that there appear to be multiple different types of problems that seem to be fundamentally different is making it difficult. Here are some example problems...
  21. C

    Find the Force and centre of pressure using double integrals

    Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=66269&stc=1&d=1391481187 The questions are on the link above. Homework Equations P = (y + 60)/10 depth (D) = y + 60 The Attempt at a Solution a) I set up the double integral: Force (F) = ∫(0 ->...
  22. S

    MHB How to Solve the Integral of Cos^2(x) Tan^3(x)?

    Stuck on this problem. Evaluate \int \cos^{2}x \, \tan^{3}x \, dx What I have so far: used the trig identity sin/cos = tan factored out a sin so I can have a even power. changed \sin^{2}x to its identity = 1/2(1 - cos2x) combined like terms and canceled out the cos \int \cos^{2}x *...
  23. H

    Integrals of motion (also First integrals)

    Hi all, Homework Statement I have got a system described by this lagrangian L(\varphi ,\psi ,\vartheta ,\dot\varphi ,\dot\psi ,\dot\vartheta )=\frac{1}{2}m(\dot\varphi^2 +\dot\psi^2 +\dot\vartheta^2 )+cos(\varphi ^2+\psi ^2). I have to find all system's integrals of motion. 2. The attempt...
  24. S

    MHB Can I Simplify Trigonometric Integrals by Taking out Constants?

    Quick question. \int sin^{4}x dx so I know: \frac{1}{2} \int 1 - 2cos2x + \frac{1}{2}(1 + cos4x)dx So here I first brought out the 1/2 because it's a constant and it's nasty. so now I have \frac{1}{4} \int 1 - 2cos2x + 1 + cos4x dx so...Just as I brought 1/2 out can I now precede to take...
  25. J

    MATLAB MatLab code for these Integrals.

    How can I write proper language for these integrals in MatLab ? Your helps really appreciated. John Mark
  26. I

    MHB Application to Improper Integrals

    Suppose that the rate that people are getting infected in an outbreak of a virus is given by y=200xe^-0.5x. How many people in total will get infected from this outbreak? So i know I'm doing it right but i keep getting a strange number… so i set up an integral of that function from 0 to...
  27. I

    MHB Improper Integrals - Comparison Test

    Hey, not too sure about what function i would compare this integral from 1 to infinity of (3x^3 -2)/(x^6 +2) dx. I also have to show that it converges. Thanks!
  28. B

    Solving Reverse Integrals: Find f(x) to Solve 1-0.1^n

    I need to find a function f(x) such that \int_{-\infty}^{100+10n} (f(x)) dx = 1-0.1^n for n=1,2,3,4,5,6...∞. How would I go about this? It must be exponential in some way I'm guessing? This is not a homework problem. I don't just want the answer. I want guidance on this type of problem...
  29. K

    Multivariable Calculus - Surface integrals

    1. Homework Statement ∫∫S xz dS where S is the boundary region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5. 2. Relevant equation∫∫Sf(x,y,z)dS = ∫∫Df(r(u,v)) * |ru χ rv|dA 3. The Attempt at a Solution I think I have broken this up into 3 surfaces. The...
  30. B

    Area element vector for parametric surface integrals

    When doing surface integrals of surfaces described parametrically, we use the area element dA = ndS = (rv x rw)dvdw Where dS is the surface area element and v and w are the parameters. I'm fine with the derivation of this (I think) but I don't understand why it's necessary to have n and dS...
  31. J

    Fundamental theorem of calculus for surface integrals?

    Hellow! A simple question: if exist the fundamental theorem of calculus for line integrals not should exist too a fundamental theorem of calculus for surface integrals? I was searching about in google but I found nothing... What do you think? Such theorem make sense?
  32. M

    Evaluate double integrals- check my work?

    Homework Statement Evaluate the iterated integrals (switch the order of integration if necessary) I just need someone to check my work. My professor gave us this practice test to help study for our final but it isn't much use if I don't know if I'm doing it correctly. I've been working...
  33. M

    Iterated integrals converted to polar

    Homework Statement ∫^{4}_{0} ∫^{√(4y-y^{2})}_{0} (x2) dx dy The attempt at a solution I'm confused on how to convert the bounds into polar coordinates. I believe x2 just becomes r2cos2θ 0≤x≤√(4y-y2) 0≤y≤4 but i don't know how to convert the bounds
  34. N

    Contour Integrals in complex analysis questions

    I am confused as to what we are obtaining when taking these contour integrals. I know that the close loop contour integral of a holomorphic function is 0. Is this analogous to the closed loop of integral of a conservative force which also gives 0? Also when I am integrating around a...
  35. alyafey22

    MHB A generalization of triple and higher power polylog integrals

    Inspired by this http://mathhelpboards.com/calculus-10/powers-polylogarithms-7998.html we look at the generalization L^m_n(p,q)=\int^1_0 \frac{\mathrm{Li}_p(x)^m\, \mathrm{Li}_q(x)^n}{x} \, dx This is NOT a tutorial. Any comments, attempts or suggestions are always welcomed.
  36. B

    Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets?

    How do I visualize \dfrac{xdy-ydx}{x^2+y^2}? In other words, if I visualize a differential forms in terms of sheets: and am aware of the subtleties of this geometric interpretation as regards integrability (i.e. contact structures and the like): then since we can interpret a...
  37. A

    Does the orientation you evaluate line integrals matter?

    If instead of evaluating the above line integral in counter-clockwise direction, I evaluate it via the clockwise direction, would that change the answer? What if I evaluate ##C_1## and ##C_3## in the counter-clockwise direction, but I evaluate ##C_2## in the clockwise direction?
  38. L

    Double Integrals: Finding the Volume of a Solid Using Polar Coordinates

    Homework Statement The plane z = 2 and the paraboloid z = 8 − 6x2 − 6y2 enclose a solid. Use polar coordinates to find the volume of this solid. Homework Equations ∫∫R f(x,y) dA = ∫βα∫ba f(rcosθ, rsinθ) r dr dθ The Attempt at a Solution z = 2, z = 8 − 6x2 − 6y2 Setting these two equal, we...
  39. I

    Double Integrals in Polar Coordinates

    Homework Statement Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid -x2 - y2 + z2 = 1 and the plane z = 2 Homework Equations r2 = x2 + y2, x = rcosθ, y = rsinθ ∫∫f(x,y)dA = ∫∫f(rcosθ,rsinθ)rdrdθ The Attempt at a Solution -x2 - y2 + 4...
  40. A

    Which of the following double integrals would correctly solve this pro

    Homework Statement Which of the following double integrals would correctly solve this problem? Homework Equations The Attempt at a Solution I obtained two sets of boundary conditions. Set 1: $$x=-\sqrt{4-y^2}\quad (for\quad x<0)\quad to\quad x=\sqrt{4-y^2}\quad...
  41. G

    Questions on conditions for calculating flux integrals?

    Conditions for calculating flux integrals? [Figured it out] If one uses Stokes' theorem and if two oriented surfaces S1 and S2 share a boundary ∂s then the flux integral of curl(F) across S1 equals the flux integral of curl(F) across S2. However, in general it won't be true that flux integral G...
  42. T

    Evaluating Volume Integrals and Divergence Theorm

    Homework Statement Evaluate the integral as either a volume integral of a surface integral, whichever is easier. \iiint \nabla .F\,d\tau over the region x^2+y^2+z^2 \leq 25, where F=(x^2+y^2+z^2)(x*i+y*j+z*k) Homework Equations \iiint \nabla .F\,d\tau =\iint F.n\,d\sigma The...
  43. MarkFL

    MHB Anh Nguyen's questions regarding indefinite integrals (integration by parts)

    Here are the questions: I have posted a link there to this thread so the OP can see my work.
  44. M

    The Classical Path, QM Path Integrals and Paths in Curved Spacetime

    "The" Classical Path, QM Path Integrals and Paths in Curved Spacetime Hey Guys! I've got an exciting question! It's been burning on my mind for years, but I think I can formulate it now. It's not so much a specific question, but rather a physical story which perhaps this thread can uncover...
  45. A

    Setting up triple integrals in different coordinates

    Homework Statement Assume that f(x,y,z) is a continuous function. Let U be the region inside the cone z=√x^2+y^2 for 2≤x≤7. Set up the intregal ∫f(x,y,z)dV over U using cartesian, spherical, and cylindrical coordinates. Homework Equations CYLINDRICAL COORDINATES x=rcosθ y=rsinθ z=z...
  46. C

    Difficult Integrals (Calculus)

    This is a forum where we chat about calculus problems that people are wondering about. I would greatly appreciate it if people also help answer some integration problems that have been nagging me for a while. Like ∫sin(sin(x))dx. Thanks Guys.
  47. MarkFL

    MHB Angelina Lopez's questions at Yahoo Answers regarding definite integrals

    Here are the questions: I have posted a link there to this thread so the OP can see my work.
  48. Q

    Given two integrals find the third

    Homework Statement The integral of f(x) from 0 to 1 is 3, and the integral of f(x) from 1 to 3 is -2. What is the integral of f(x) from -3 to 3? Homework Equations FTC. The Attempt at a Solution From the equations given I know: F(1) - F(0) = 3, and F(3) - F(1) = -2...
  49. S

    MHB Evaluating definite integrals via substitution.

    Can someone make sure I'm on the right track with this problem? I'm a little confused because I thought that when you make a substitution you update the limits and get better numbers to work with when you plug them in the function in the end...Yet, it seems like I almost got worse numbers to...
  50. T

    How to Determine Line and Surface Integrals with Rectangular Boundaries

    Homework Statement Consider a vector A = (2x-y)i + (yz^2)j + (y^2z)k. S is a flat surface area of a rectangle bounded by the lines x = +-1 and y = +-2 and C is its rectangular boundary in the x-y plane. Determine the line integral ∫A.dr and its surface integral ∫(∇xA).n dS Homework...
Back
Top