What is Integral: 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. karush

    MHB 7.3.5 Integral with trig substitution

    $\textsf{Evaluate the integral}$ $$I=\displaystyle\int\frac{x^2}{\sqrt{9-x^2}}$$ $\textit{from the common Integrals Table we have}$ $$\displaystyle I=\int\frac{u^2}{\sqrt{u^2-a^2}} \, du =\frac{u}{2}\sqrt{u^2-a^2}+\frac{a^2}{2} \ln\left|u+\sqrt{u^2-a^2}\right|+C$$...
  2. Javier Lopez

    Elliptic integral of the third order for magnet calculus

    I have the first and second orders that I use in a magnetic simulator, but i need the thirth also to do also with magnetic cylinders accordingly paper: Do anybody have it in any code? I should pass to C++
  3. K

    Using the Integral Test to Show Sum is Less Than pi/2

    Homework Statement Use the integral test to show that the sum of the series gif.latex ##\sum_{n=1}^\infty \dfrac{1}{1+n^2}## is smaller than pi/2. Homework EquationsThe Attempt at a Solution I know that the series converges, and the integral converges to pi/4. As far as I´ve understood...
  4. T

    How Do You Solve a Complex Integral Using Cauchy-Goursat's Theorem?

    Homework Statement ##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks! Homework Equations Cauchy-Goursat's Theorem The Attempt at a Solution Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...
  5. R

    Integral of Acceleration with respect to time

    Homework Statement Acceleration is defined as the second derivative of position with respect to time: a = d2x/dt2. Integrate this equation with respect to time to show that position can be expressed as x(t) = 0.5at2+v0t+x0, where v0 and x0 are the initial position and velocity (i.e., the...
  6. T

    How to evaluate a surface integral with three points?

    Homework Statement Let G=x^2i+xyj+zk And let S be the surface with points connecting (0,0,0) , (1,1,0) and (2,2,2) Find ∬GdS. (over S) Homework EquationsThe Attempt at a Solution I parametrised the surface and found 0=2x-2y. I’m not sure if this is correct. And I’m also uncertain about...
  7. 0

    I don't understand this integral

    What did the teacher meant with this: $$\int_{a}^{b} f(t)i + g(t)k dt $$ The two functions, a and b are all given. What is it to integrate a vector? From analytical geometry I know that something in the form of i + j + k is a vector.
  8. A

    Show that the integral converges

    Homework Statement (FYI It's from an Real Analysis class.) Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent. Homework Equations I know that for an integral to be convergent, it means that : $$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.I can also use the...
  9. M

    How Should Exponential Terms Be Integrated in Fourier Transforms?

    Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can...
  10. learning_physica

    Work: Dot Product and Integral?

    I’m having trouble understanding the relationship between how work is both a dot product and integral. I know that work equals F • D and also the integral of F(x): the area under the curve of F and D. However, let’s say that I have a force vector <3,4> and a displacement vector of <3,0>. The...
  11. L

    Electric Potential Integral Question Confused

    Homework Statement Homework Equations E=KQ/R^2 The Attempt at a Solution I'm kinda confused at what the question is asked. It is in terms of x, but I thought the integral for potential is V=int(Edr)? Also, should it be integration starting from infinity? Why is the integration from -2 to 3...
  12. T

    MHB Definite integral with Riemann sums

    Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...
  13. S

    MHB Problems for Cauchy Integral Formula

    Hello everyone! I am currently stuck at the two type of questions below, because I am not really sure what method should be used to calculate these question... Could you give me a hint how to do these questions? :(
  14. S

    I QFT Path Integral: Confusions on Left & Right Most Terms & Vacuum State

    Hello! I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N exp(i\delta t \int d^3x L[\Phi_j,\partial_t \Phi_j])$$ What happens when we have the left and right...
  15. M

    Calculate the given surface integral [Mathematical physics]

    Homework Statement Calculate \int_{S} \vec{F} \cdot d\vec{S} where \vec{F} = z \hat{z} - \frac{x\hat{x} + y \hat{y} }{ x^2 + y^2 } And S is part of the Ellipsoid x^2 + y^2 + 2z^2 = 4 , z > 0 and the normal directed such that \vec{n} \cdot \hat{z} > 0 Homework Equations All the...
  16. A

    MHB How to integral legendre polynomial

    Question \int_{-1}^{1} cos(x) P_{n}(x)\,dx ____________________________________________________________________________________________ my think (maybe incorrect) \int_{-1}^{1} cos(x) P_{n}(x)\,dx \frac{1}{2^nn!}\int_{-1}^{1} cos(x) \frac{d^n}{dx^n}(x^2-1)^n\,dx This is rodrigues formula by...
  17. K

    Maximum value of this integral

    Homework Statement Find a > 0 so the integral int(exp(-ax)*cosx)dx from 0 to inf get as high value as possible. The Attempt at a Solution My way of solving this is to plot the integrand, i.e. exp(-ax)*cosx and check for different values of a. The larger a is, the smaller the area under the...
  18. T

    MHB Basic definite integral question

    Hey, I've got this problem I've been doing, but I'm not sure if my approach is right. My textbook has pretty much less than a paragraph on this sort of stuff. My thinking was that since an integral is a sum, in order to get the range from 0 to 8, we should just be able to add or subtract the...
  19. Y

    MHB Improper integral from 1 to infinity

    Hello everyone, I am stuck on this homework problem. I got up to (ln (b / (b+1) - ln 1 / (1+1) ) but I'm not sure how to go to the red boxed step where they have (1 - 1 / (b+1) ) if anyone can figure it out Id really appreciate it. thank you very much.
  20. K

    How Do Zeroes of a Function Relate to Its Integral?

    Homework Statement You are given the function f(x)=3x^2-4x-8 a) Find the values of a. Explain the answers using the function. Homework EquationsThe Attempt at a Solution a^3-2*a^2-8*a=0 a=-2 v a=0 v a=4 I found the answers, but I don't know how to explain my answers by using the function...
  21. gibberingmouther

    Line Integral for Electromagnetic Force

    http://web.mit.edu/sahughes/www/8.022/lec01.pdf So I'm trying to understand how to get from F = ∫[(Q*λ)*dL*r]/(r^2) to F=∫q*λ*[(xx+ay)/(a^2+x^2)^(3/2)]*dx Like I don't understand why the x and y components of r are negative, or why "The horizontal r component is obviously zero: for every...
  22. S

    Double Integral: How to Evaluate a Double Integral over a Pentagonal Region

    Homework Statement Evaluate ##\int\int_{R} (x+2)(y+1) \; dx \; dy## where ##R## is the pentagon with vertices ##(\pm 1,0)##, ##(\pm 2,1)## and ##(0,2)##. Homework EquationsThe Attempt at a Solution After drawing ##R## I split ##R## into two sections ##R_1## (left half) and ##R_2## (right half)...
  23. M

    [Mathematical physics] - Integral problem

    Homework Statement Calculate the integral \int_{S} (\frac{A}{r^2}\hat{r} + B\hat{z}) \cdot d\vec{S} Where S is the sphere with r = a. 2. The attempt at a solution I have no clue how to solve this problem. I have thought of introducing spherical coordinates and somehow finding a connection...
  24. beefbrisket

    I Sign mistake when computing integral with differential forms

    The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...
  25. A

    I Learning the theory of the n-dimensional Riemann integral

    I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...
  26. Bishal Banjara

    How to reduce the integral equation for light deflection?

    1. At pg.212, Hartle book (2003) writes equation 9.81 as an approximation of 9.80, directly. 2. $$ΔΦ=\int_0^{w_1}\frac{(1+\frac{M}{b}w)}{(1+\frac{2M}{b}w-w^2)^\frac{1}{2}}dw$$ equation(9.80) $$ΔΦ≈\pi+4M/b$$...
  27. D

    I Can Cauchy's Integral Formula be Used for Non-Analytic Functions?

    Hi. I am working my way through some complex analysis notes(from a physics course). I have just covered Cauchy's theorem which basically states that the integral over a closed contour of an analytic function is zero. this is then used to show that contours of analytic functions can be deformed...
  28. T

    I Intuition - Cauchy integral theorem

    So folks, I'm learning complex analysis right now and I've come across one thing that simply fails to enter my mind: the Cauchy Integral Theorem, or the Cauchy-Goursat Theorem. It says that, if a function is analytic in a certain (simply connected) domain, then the contour integral over a simple...
  29. S

    MHB Does the Norm of a Linear Integral Operator Equal Its Spectral Radius?

    Hello A simple question. I have a linear integral operator (self-adjoint) $$(Kx)(t)=\int_{a}^{b} \, k(t,s)\,x(s)\,ds$$ where $k$ is the kernel. Can I say that its norm (I believe in $L^2$) equals the spectral radius of $K?$ Thanks! Sarah
  30. A

    How to solve this electrostatic potential integral equation with Python

    Hi! I would like solve this kind of relation: \phi = \int_0^r \phi (r') 4 \pi r'dr' But I don't know how to proceed... Can you advise me ? Thank's in advance !
  31. S

    What's a good FREE font for integral signs?

    I like use OpenOffice to do mathematics documents (yes, not as good as LaTex, but it's not as much of a hassle), but I find the fonts that I currently have don't have a good set of good-looking integral signs, so I looking for a FREE font that has a good set of such signs. A small search for...
  32. Entertainment Unit

    Integral of 1/(a sin^2 x + b sin x cos x + c cos^2 x)

    Homework Statement If ##a \neq 0##, evaluate the integral $$\int \frac {dx} {a~\sin^2~x + b~\sin~x~\cos~x + c~\cos^2~x}$$ (Hint: Make the substitution ##u = \tan x## and consider separately the cases where ##b^2 - 4ac## is positive, zero, or negative.) The Attempt at a Solution $$\int \frac...
  33. lfdahl

    MHB Is the definite integral ∫ [arcsin(1/x)-1/x]of indeterminate form?

    Is the definite integral $$\int_{1}^{\infty}\left(\arcsin \left(\frac{1}{x}\right)-\frac{1}{x} \right)\,dx$$ of indeterminate form or not? Prove your statement.
  34. E

    I How Can I Solve This Complex Integral Evaluation Problem?

    Hi, I have the following integral that I want to evaluate: \int_0^{\infty}y\,e^{-y\left[(z+1)(K-1)+1\right]}Ei\left(y_2(K-1)\right)\,dy In the table of integrals there is a similar integral in the form \int_0^{\infty}x^{v-1}\,e^{-\mu...
  35. Math Amateur

    MHB Irreducible Elements and Maximal Ideals in Integral Domains ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to formulate a proof of Proposition 4.3.5 Part (iii)... ... Proposition 4.3.5 reads as...
  36. M

    B Triple integral in spherical coordinates.

    While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...
  37. E

    I Closed-form solution for a triple integral

    Hello all, I need to evaluate the following 3-dimensional integral in closed-form (if possible) \int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min(x_2,\,y_1(z-\frac{x_2}{y_2}))\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1 where ##z## is real positive number, and...
  38. E

    I Closed-form solution for an integral

    Hello all, Is there a closed form solution for the following integral \int_0^z\frac{1}{1+z-x}\frac{1}{(1+x)^K}\,dx for a positive integer ##K\geq 1##, and ##z\geq 0##? I searched the table of integrals, but couldn't find something similar. Thanks in advance for any hint
  39. Krushnaraj Pandya

    Problem involving a derivative under the integral sign

    Homework Statement if ## f(x) ={\int_{\frac{\pi^2}{16}}^{x^2}} \frac {\cos x \cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz## then find ## f'(\pi)## 2. The given solution Differentiating both sides w.r.t x ##f'(x) = {-\sin x {\int_{\frac{\pi^2}{16}}^{x^2}} \frac{\cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz }+{...
  40. Krushnaraj Pandya

    Definite trigonometric integral

    Homework Statement solve ##\int_0^1 x^6 \arcsin{x} dx##
  41. Krushnaraj Pandya

    Definite trigonometric integral using properties

    Homework Statement If ## I_n = \int_0^\frac {\pi}{4} \sec^n x dx## then find ## I_{10} - \frac {8}{9} I_8## 2. The attempt at a solution this should be solvable by reduction formulae but since it'd be longer I wanted to know if there was a way to do it using mostly properties of indefinite...
  42. Krushnaraj Pandya

    Definite integral using only properties

    Homework Statement evaluate the following definite integral with limits 0 to 1## ∫log(sin(πx/2)) dx ## 2. The attempt at a solution I used ##∫f(x) = ∫f(a+b-x)## to get ## I=∫log(cos(πx/2))## with the same limits. Adding them and using ##log(m)+log(n)=log(mn)## and ##2sinxcosx=sin2x## I got...
  43. Mr Davis 97

    I Differentiating under the integral sign

    I am looking at a solution to an integral using differentiation under the integral sign. So let ##\displaystyle f(t) = \frac{\log (tx+1)}{x^2+1}##. Then, through calculation, ##\displaystyle f'(t) = \frac{\pi t + 2 \log (2) - 4 \log (t+1)}{4(1+t^2)}##. The solution immediately goes to say that...
  44. E

    MATLAB Evaluating Nested Integral in MATLAB for General K

    Hello, How can I evaluate the following nested integral in MATLAB for a general value of ##K## {\int\limits_{u_1=0}^{\gamma}\int\limits_{u_2=0}^{\gamma-U_1}\cdots \int\limits_{u_{K}=0}^{\gamma-\sum_{k=1}^{K-1}U_k}}f_{U_1}(u_1)f_{U_2}(u_2)\cdots f_{U_{K}}(u_{K})\,du_1du_2\cdots du_{K} where...
  45. SamRoss

    B Relativistic Energy: Change of Consts. of Integration

    In this super short video of the derivation of the relativistic kinetic energy, , I'm just stuck on one thing. Around 1:00 minute in, the constants of integration change from 0 to pv when the integration changes from dx to dv. Where does the pv come from? Thanks!
  46. W

    Drawing conclusions by looking at integral

    Homework Statement I have the following expression $$I = \int_{-\infty}^{0} f_p(p) \ \big[ pf_x(a - \frac{p}{m}t) \big] dp + \int_{0}^{\infty} f_p(p) \ \big[ pf_x(a - \frac{p}{m}t) \big] dp$$ where ##f_p## and ##f_x## are normalised distributions. In particular, ##f_x## is symmetric about...
  47. mishima

    Boas 4.12.18, 2nd Derivatives of Imp. Multivariable Integral

    Homework Statement Show that u(x, y) = y/π ∫-∞∞ f(t) dt / ((x - t)2+y2) satisfies uxx + uyy = 0. Homework Equations Leibniz' Rule The Attempt at a Solution I'm not even sure Leibniz' Rule can be applied here since there seems to be a discontinuity in the integrand when x=t and y=0. When I...
  48. FallenApple

    Bounds Integral of x times arcsine

    Homework Statement Prove the integral of x*arcsine(x) from 1/2 to 1 is bounded between pi/16 and 3*pi/16 Homework EquationsThe Attempt at a Solution Not sure what to bound with. Do we use Squeeze Theorem?
  49. Mr Davis 97

    Mathematica Trying to calculate an integral

    This is actually a WolframAlpha question, but I suppose someone conversant in mathematica could give me an answer. How in Mathematica could I compute ##\displaystyle \int_0^1 \left( \prod_{r=1}^3 (x+r)\right) \left(1+x \sum_{r=1}^3 \frac{1}{x+r} \right) ~ dx##. I tried int (Product[x+r, {r...
  50. B

    Use triple integral to find center of mass

    Homework Statement Find the centre of mass of a uniform hemispherical shell of inner radius a and outer radius b. Homework Equations ##r_{CoM} = \sum \frac{m\vec{r}}{m}## The Attempt at a Solution Using ##x(r,\theta,\phi)## for coordinates...
Back
Top