Integral Definition and 188 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.

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  1. E

    Evaluating cosine function from ##-\infty## to ##\infty##

    Hi, I have some question about evaluating a cosine function from ##-\infty## to ##\infty##. I saw for a cosine function evaluate from ##-\infty## to ##\infty## I can change the limits from 0 to ##\infty##. I have a idea why, but I can't convince myself, furthermore, is it always the case no...
  2. C

    Prove limit comparison test for Integrals

    Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
  3. Johan_S

    A Yet another cross-product integral

    I am trying to integrate a cross product and I wonder if the following is true. It does not feel like it is true but it would be very nice if it was since otherwise I have a problem with the signs... This is my first time posting here, so I just pasted in the LaTeX code and hope that it is...
  4. patric44

    I Has the Fermi-Dirac Integral been solved?

    hi guys I have a question about whether or not the Fermi-Dirac Integral has Been solved, because i found a formula on Wikipedia that relates the Fermi-Dirac integral with the polylogarithm function: $$F_{j}(x) = -Li_{j+1}(-e^{x})$$ and in some recent papers they claim that no analytical...
  5. A

    I Index and bound shift in converting a sum into integral

    Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted? $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$
  6. Saracen Rue

    B Integrate 1/x from -1 to 1

    I've always been taught that the indefinite integral of ##\frac{1}{x}## is ##\ln(|x|)##. Extending this to definite integrals, particularly over limits involving negative values, should work just like any other integral: $$\int_{-1}^{1} \frac {1} {x} dx = \ln(|-1|) - \ln(|1|) = \ln(1) - \ln(1)...
  7. E

    Integration ## f(\theta, \phi) = \frac{sin \theta}{4\pi}##

    Hi, I have this formula ## f(\theta, \phi) = \frac{sin \theta}{4\pi}## I have this statement that say if I integrate this formula above on a sphere then p = 1. what does integrate on a sphere means? I know ##\int_0^{2\pi} ## is used for the circle.
  8. redtree

    I Integrating with the Dirac delta distribution

    Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...
  9. E

    ##\int_0^1 F_x(x,0)dx + \int_0^1 F_y(1,y)dy##

    Hi, This is the first time I see this kind of integral. I'm not sure how to resolve it. ## \int_0^1 F \cdot dr ## ## \int_0^1 F_x(x,0)dx + \int_0^1 F_y(1,y)dy ## ##F = (y,2x)## I don't know the values of ## F_x(x,0) ## and ## F_y(1,y)##
  10. E

    Solve this integral involving a quadratic and linear air resistance equation

    Hi, I'm trying to solve this integral and then isolate V, but I can't get the right answer. I don't know where is my errors. I probably muffed the integral. ##-bv -cv² = m\frac {dv}{dt}## ## \int_0^t dt = - m \int_{Vo}^v \frac {dv}{bv+cv^2} ## I get this after the integration ##t =...
  11. K

    Calculating crossproduct integral, Parametrization

    i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right? The surface occupied by the cylinder can be described by all vectors ##\vec x =(x,y,z)## so that...
  12. tworitdash

    A Spectral domain double integral with singularities

    The integral looks like Y_{mut, mn} = -j^{m+n}nm \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{2 ab (k^2 - k_x^2) \sin^2(\frac{k_yb}{2}) \cos^2(\frac{k_xa}{2})}{\omega \mu k_z (\frac{k_yb}{2})^2 [(n\pi)^2 - (k_xa)^2][(m\pi)^2 - (k_xa)^2]} dk_x dk_y Here, k_z = -1j...
  13. tworitdash

    A Spatial Fourier transform of a Bessel function multiplied with a sinusoidal function

    I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m +...
  14. G

    Determining a centroid

    Summary:: I'm solving an exercise. I have the following center of gravity problem: Having the function Y(x)=96,4*x(100-x) cm, where X is the horizontal axis and Y is the vertical axis, ranged between the interval (0, 93,7) cm. Determine: a) Area bounded by this function, axis X and the line...
  15. B

    A Can this difficult Gaussian integral be done analytically?

    Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...
  16. Michael_0039

    Find the volume of the solid formed by the rotation around the y=0

    Hi, I find this.... Please tell me your opinion on this. Thanks.
  17. Michael_0039

    Ιntegral calculation : (sin(x))^4 * (cos(x))^6

    Summary: Ιntegral calculation : (sin(x))^4 * (cos(x))^6 Hi all, I tried to solve it, but I got stuck. An advice from my professor is to set: x=arctan(t) Τhanks.
  18. C

    Differential Integration Problem

    Attempt at solution: Writing the chain rule for ## E(V,T) ##: ## dE = \frac{\partial E}{\partial T}dT + \frac{\partial E}{\partial V}dV ## Then, integrating the differential: ## \int{ dE } = \int{ \frac{\partial E}{\partial T}dT } + \int{ \frac{\partial E}{\partial V}dV } ## If I put the...
  19. SamRoss

    I Why wasn't this symbol "swapped"?

    In a certain derivation, the author begins with $${g(-t)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(\omega)}e^{-i\omega t}d\omega$$ and then says he will replace ##t## with ##\omega## and ##\omega## with ##t##. He then writes $${g(-\omega)=}\frac 1 {2\pi}\int_{-\infty}^\infty {G(t)}e^{-it\omega...
  20. fazekasgergely

    Infinite series to calculate integrals

    For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
  21. SamRoss

    B Justification for cancelling dx in an integral

    In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))... ## \int_0^\phi \frac {d} {dx} f(x) dx =...
  22. S

    Analytical solution of the Photon Diffusion Equation

    Homework Statement Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique. Homework Equations Starting equation integrated over t: int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1) Result...
  23. CCMarie

    A Multi-variable function depending on the Heaviside function

    How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...
  24. matai

    Integral for the linear speed of the Earth

    I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the earth rotates at 6.963448857E-4 revs/min...
  25. SamRoss

    B Can you help me see why these integrals are the same?

    I am reading "Inside Interesting Integrals" by Paul Nahin. Around pg. 59, he goes through a lengthy explanation of how to do the definite integral from 0 to infinity of ∫1/(x4+1)dx. However, he then simply writes down that this integral is equal to ∫x2/(x4+1)dx with the same limits. Now, it's...
  26. V

    B Tips for solving an Integral (angles in a metric of a spherical, 2-D surface)

    Hello. I ask for solution help from the integral below, where y and x represent angles in a metric of a spherical, 2-D surface. He was studying how to obtain the geodesic curves on the spherical surface, the sphere of radius r = 1, to simplify. The integral is the end result. It is enough, now...
  27. CivilSigma

    Find the PDF in terms of another variable

    Homework Statement For $$f_x(x)=4x^3 ; 0 \leq x \leq 1$$ Find the PDF for $$ Y < y=x^2$$ The Attempt at a Solution So, we take the domain on x to be: $$0\leq x \leq \sqrt y$$ and integrate: $$ \int_0^{\sqrt y} f_x(x) dx = \int_0^{\sqrt y} 4x^3 dx$$ Do we integrate with respect to x or y...
  28. nomadreid

    I |Li(x) - pi(x)| goes to 0 under RH?

    Extremely quick question: According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to |Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c. Am I correct that then c goes to 0 as x goes to infinity? Does any expression exist (yet) for c? Thanks.
  29. CivilSigma

    Auto-correlation Integral

    Homework Statement I am computing the auto correlation and spectral density functions of the following signal: $$f(t)=Ae^{-ct}sin(\omega t)$$ $$AutoCorrelation = R_x(\tau) = \int_{-\infty}^{\infty} f(x)f(x+\tau) \cdot \frac{1}{T} dx$$ $$SpectralDensity = S_x(\omega) = \frac{1}{2\pi}...
  30. Zack K

    Electric field of a curved rod

    Homework Statement A rod of charged -Q is curved from the x-axis to angle ##\alpha##. The rod is a distance R from the origin (I will have a picture uploaded). What is the electric field of the charge in terms of it's x and y components at the origin? k is ##\frac {1} {4\pi \epsilon_0}##...
  31. A

    I Integration being unchanged after rotation

    This question is about the general 1 loop correction to the propagator in QFT (this is actually not important for this question). Let's say we have an integral over an integration variable x, and this x ranges from ##-\infty## to ##\infty##. If we look at this integration contour in the complex...
  32. N

    Python How to plot integration equation using Python?

    I have a few of integration equations and need to convert it into Python. The problem is when I tried to plot a graph according to the equation, some of the plot is not same with the original one. The first equation is the error probability of authentication in normal operation: cond equation...
  33. H

    ##\int (\sin x + 2\cos x)^3\,dx##

    Homework Statement $$\int (sinx + 2cos x)^3dx$$ Homework Equations The Attempt at a Solution $$\int (sinx + 2cos x)^3dx$$ $$\int (sinx + 2cos x)((sinx + 2cos x)^2dx)$$ $$\int (sinx + 2cos x)(1 + 3cos^2x+2sin2x)dx$$ How to do this in simpler way?
  34. Adesh

    I How to convert the limit of a series into an integral?

    If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral. Thank you.
  35. 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...
  36. K

    I need some help with integrals

    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 Equations The 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...
  37. SamRoss

    B Change of constants of integration for relativistic energy

    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!
  38. E

    A How to solve this trigonometric integral?

    Hello everyone Can someone help me out solving this integral: \begin{equation} S_T(\omega)=\frac{2k_BT^2g}{4\pi^2c^2}\int_0^{\infty}\frac{sin^2(kl)}{k^2l^2}\frac{k^2}{D^2k^4+\omega^2}dk \end{equation} Where $$D=g/c$$ According to this paper https://doi.org/10.1103/PhysRevB.13.556. The...
  39. T

    A simple case of translation invariance of Riemann integrals

    Homework Statement Show that \int_{A} 1 = \int_{T(A)} 1 given A is an arbitrary region in R^n (not necessarily a rectangle) and T is a translation in R^n. Homework Equations Normally we find Riemann integrals by creating a rectangle R that includes A and set the function to be zero when x...
  40. O

    Time derivative of gravity due to acceleration

    Homework Statement We have the equation for gravity due to the acceleration a = -GM/r2, calculate velocity and position dependent on time and show that v/x = √2GM/r03⋅(r/r0-1) Homework Equations x(t = 0) = x0 and v(t = 0) = 0 The Attempt at a Solution v = -GM∫1/r2 dt v = dr/dt v2 = -GM∫1/r2...
  41. W

    Maple Computing Numerical Integrals with Maple

    Hi all, I am new to the Maplesoft software and have been experiencing trouble computing numerical integrals. I defined a few mathematical functions in terms of a few variables like so: I then used "subs" to input values to anything that isn't already a defined constant (like ##\hbar,\pi## and...
  42. W

    Numerical/Analytical Solution to a Complex Integral

    Homework Statement I have the following integral I wish to solve (preferably analytically): $$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$ where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...
  43. H

    How to solve this integral of an absolute function?

    Homework Statement Homework Equations The Attempt at a Solution I think the answer for number 1 , graph somewhat like this I get trouble for 2, 3, etc I (k) = ##\int_{-1}^{1} f(x) dx ## f(x) = ## \mid x^2 - k^2 \mid## 2) k < 1 for negative side ##\int_{-1}^{-k} (x^2 - k^2) dx +...
  44. hugo_faurand

    B Calculate the expression of the antiderivative

    Hello everyone ! I've started to work on integral and I wonder if it's possible to calculate the expression of the antiderivative with the expression of the "integrand"1 rather than use a table with the function and its antiderivative. Thank you in advance ! 1( I'm french and I d'ont know the...
  45. SemM

    I Understanding an integral

    Hi, I have the following integral which I am not confident on how to interpret (solve): \begin{equation} \alpha \bigg( \int_0^L [\frac{d^3}{dx^3}\phi] \psi dx - \int_0^L [\frac{d^3}{dx^3}\psi] \phi dx \bigg) \end{equation} at this stage, I am not sure which rule to use to solve each of the...
  46. J

    A Maximization Problem

    Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2} $$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...
  47. nightingale123

    I N-th dimensional Riemann integral

    Hello I have a question regarding something we wrote in class today. Let ##A## be a bounded subset of ##R^n##, let ##f,g:A\to \mathbb{R}## be integrable functions on A. ##a)## if ## A## has a volume and ##\forall x \in A :m\leq f(x) \leq M## then ##mV(A)\leq \int_{A}f(x)\leq MV(A)## this...
  48. GaussianSurface

    How can I find this displacement?

    Homework Statement Your task is to estimate how far an object traveled during the time interval 0≤t≤8, but you only have the following data about the velocity of the object. *First image You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue...
  49. GaussianSurface

    Calculating distance from speed

    Homework Statement The speed of a runner increased during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she travelled during these three seconds. It follows the image's square. Homework...
  50. W

    Writing integrals in terms of the error function

    Homework Statement I have the following integral, $$\frac{1}{\sigma \sqrt{2\pi} t} \int_{-\infty}^{0} \exp[\frac{-1}{2\sigma ^2} (\frac{x-x_0}{t} - p_0)^2]dx$$ that I wish to write in terms of the error function, $$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-g^2}dg$$ However, I can't seem...
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