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.

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  1. V

    B How to set up an integral to integrate over a sine wave?

    How do I setup an integral to integrate over the following equation: V(t) = 1/(R*C) integral to t Vin(t) dt This is the capacitor voltage over time formula. I want to integrate over a sine wave from 9 to 81 degrees. Frequency of 120Hz, amplitude of 120V. The formula I used in wolframalpha is...
  2. CivilSigma

    Integral of a Recursive Function

    Homework Statement Find the following: $$ \int_0^\inf \frac{x^{a+1}e^{-x/\delta}}{\delta^{a+1}\Gamma(a+1)} dx; \, a > 1 , \delta >0 , 0 \leq x \leq \inf$$ Homework Equations - The Attempt at a Solution The numerator in the integral is constant, so it can be taken outside the integral. I then...
  3. weirdoguy

    Discrete K-G eq. solutions - problem with integral

    I'm reading Klaubers QFT book and I stuck with his derivation of Hamiltonian of scalar field on page 53. To derive it one needs to deal with integrals like this: $$\int\dot{\phi}\dot{\phi}^\dagger d^3x$$ He is using discrete plane-wave solutions and after plugging them in, we end up with...
  4. Burhan Uddin

    Calculating the work done using a line integral

    Homework Statement a) A point charge + q is placed at the origin. By explicitly calculating the relevant line integral, determine how much external work must be done to bring another point charge + q from infinity to the point r2= aŷ ? Consider the difference between external work and work...
  5. 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}##...
  6. F

    B Integrating sec(x): Understanding the Answer

    Hello. I integrated ##\sec(x) ## and got an answer. I differentiated it to verify it and it came out well. Later when I was looking in a table of integrals I saw the solution ## \ln| \sec(x) +\tan(x) | + c##. This was completely different than my solution. I do not think I made a mistake...
  7. Hawkingo

    How to find the limits of a volume integral?

    Homework Statement If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##. The Attempt at a Solution From gauss divergence theorem we know ##\int...
  8. 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...
  9. Robin04

    How to calculate this integral?

    Homework Statement I have a problem with this integral: ##\int_{x_0}^x \frac{dx}{\sqrt{2-2\cos{x}}}## Homework EquationsThe Attempt at a Solution I came across this while reading a book and the author says that this can be written in the form ##\int_{x_0}^x \frac{dx}{2\sin{\frac{x}{2}}}##. I...
  10. K

    Integral of a differential form

    Homework Statement Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$ \int_{\mathbb{R}^n} d\omega = 0. $$ Homework Equations [/B] $$\oint_{\partial K} \omega = \int_K d\omega$$ The Attempt at a Solution...
  11. opus

    Trig Integral Homework: Solving \int_0^{π/8}sin^2(x)cos^2(x)

    Homework Statement ##\int_0^{π/8}sin^2(x)cos^2(x)## Homework EquationsThe Attempt at a Solution Please see my attached work to see the train of thought. I've tried this thing about 100 times and still can't get the correct solution. I don't know if it's in the anti derivative evaluations of...
  12. Q

    Cylindrical Coordinates: Line Integral Of Electrostatic Field

    Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...
  13. B

    Quantum mechanical integral equation problem

    Homework Statement The question is; for a qunatum mechanical particle, Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h] in here, x0, p0 and h are constants, so, Homework Equations what are the <x>; expetation value, and <P>;expectation value of momentum ? The Attempt at a Solution , [/B]...
  14. B

    A Quantum mechanical integral equation problem

    The question is; for a qunatum mechanical particle, Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h] in here, x0, p0 and h are constants, so, what are the <x> and <P> ?
  15. opus

    I Volume of revolution -- Why use this integral?

    My question is, why is the circled integral the chosen integral for this case? My thoughts are that we don't just use ##\int_0^1e^{-x}## because we need to make this two dimensional area into a three dimensional volume by doing 360 degrees of rotation. This would correspond to ##2πr##. ##2π## is...
  16. V

    MHB Solving Limits of Integrals: Advice & Tips

    I tried to use integration by parts. I took f(x)=arctan(x) => f'(x)= 1/x^2+1 g'(x)=cos(nx) => g(x)= sin(nx)/n So I get sin(nx)/n * arctan(x) - integral from 0 to 1 from sin(nx)/n(x^2+1) How to continue ? I'm always getting stuck with this kind of exercises ( limits of integrals ) because I don't...
  17. R

    I Finding the value of the integral on this t-shirt

    Hello all, I was watching some random videos on YouTube and found one of Dr. Ben Goertzel (a leading expert in the field of AI) and he was wearing the t shirt you see below. It caught my attention. I tried working it out, but I had no luck as it got messy fast. I then tried throwing it into...
  18. barryj

    A Hint as to how to solve an integral differential equation

    I know how to solve a differential equation using Eulers method but what if the equation has an integral part? i.e. a RLC electrical circuit. Vsource = iR + L di/dt + (1/C)int i dt can this be done? a link to how to solve this would be helpful.
  19. G

    MHB How Do You Solve an Integral with Two Variables?

    Hey there, trying to figure out how to solve this integral (see picture). I've never seen an integral written in this way before. I've tried to integrate the x-part first and then the y-part and vice versa but they both gave the wrong results.
  20. N

    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...
  21. cookiemnstr510510

    Evaluate the integral: ∫csc((v-π)/2)cot((v-π)/2)

    Homework Statement Evaluate the integral: ∫csc((v-π)/2)cot((v-π)/2) Homework Equations cscx=1/sinx cot=sinx/cosx The Attempt at a Solution I first turned csc and cot into the above "relevant equations" ∫ (1/sin(##\frac{v-π}{2}##)(##\frac{sin(v-π)/2}{cos(v-π)/2}##)=∫cos-1((v-π)/2) then...
  22. F

    Simplifying integral question....

    Homework Statement I have solve the integral for:to be: But I cannot figure out how to simplify the answer to the form shown above. Homework Equations The Attempt at a Solution Here is my progress so far: Any help would be appreciated
  23. L

    B Line Integral, Dot Product Confusion

    From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
  24. Boltzman Oscillation

    What is the integral of this Gaussian distribution?

    Homework Statement Find A in p(x) = Aexp(-λ(x-a)^2) by using the equation 1 = ∫ p(x)dxHomework Equations 1 = ∫p(x)dx The Attempt at a Solution I expand the power of the exponential and then extract the constant exponential to get: Aexp(λa^2) ∫exp(-λx^2)exp(2aλx)dx I don't know how to...
  25. Michael Santos

    The indefinite integral and its "argument"

    Homework Statement The indefinite integral $$\int \, $$ and it's argument. The indefinite integral has a function of e.g ## \cos (x^2) \ ## or ## \ e^{tan (x)} \ ## If the argument of ## \cos (x^2) \ ## is ## \ x^2 \ ## then the argument of ## \ e^{tan(x)} \ ## is ## \ x \ ## or ## \ tan (x) \...
  26. Y

    Integral Test: Analyzing $$\sum_{n=3}^{\infty} \frac{1}{n^2-4}$$

    Homework Statement Use the integral test to determine whether $$\sum_{n=3}^{\infty} \frac{1}{n^2-4}$$ converges or diverges.The Attempt at a Solution [/B] Taking the integral we have $$\int_{}^{\infty}\frac{dn}{n^2-4}$$ Note: Mary Baos text is having us write the integrals without lower bounds...
  27. ChristinaMaria

    Improper integral with substitution

    Hi! I am trying to solve problems from previous exams to prepare for my own. In this problem I am supposed to find the improper integral by substituting one of the "elements", but I don't understand how to get from one given step to the next. Homework Statement Solve the integral by...
  28. H

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

    Homework Statement $$\int (sinx + 2cos x)^3dx$$ Homework EquationsThe 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?
  29. N

    Deriving the ∫dω1 integral in 2nd-order polarization

    Homework Statement Starting with the second order polarization in the time domain: (1) I am trying derive the frequency domain form: (2) Multiple sources give essentially the same formula with the same integral, I have obtained the particular ones in here from those lecture notes. My...
  30. Boltzman Oscillation

    How would I perform this surface integral?

    Homework Statement ∫∫ F ⋅ ndτ over the spherical region x^2 + y^2 + z^2 = 25 given F = r^3 r i already converted the cartesian coordinates to spherical in FHomework Equations n = r[/B]The Attempt at a Solution I know I can plug in F into the equation and then dot it with r to get the...
  31. 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.
  32. D

    Step Change in a proportional plus integral controller

    Homework Statement Figure 5 shows the input and output wave forms for a proportional plus integral controller. State: (i) the controllers proportional gain (ii) the controllers integral action time The attachment below is a copy and paste of the input and output wave forms I am given...
  33. D

    I A surface integral over infinite space

    Hi. If a function f is normalizable ,ie f→0 as | x | → infinity or r→ infinity then I presume the following surface integral f dS over infinite space is zero ? But I thought about this again and it seems like a case of zero x infinity. The function is zero at the infinite surface but the area...
  34. Hiero

    Integral of a normal distribution

    Homework Statement (Scroll to bottom for the true question) Suppose we are to find the integral from -∞ to +∞ of (let’s just say) e-x2dx Homework Equations ∫∫f(x)g(y)dxdy = (∫f(x)dx)(∫g(y)dy) The Attempt at a Solution We can square the integral we want to solve for then use my relevant...
  35. CptXray

    Finding integral curves of a vector field

    Homework Statement For a vector field $$\begin{equation} X:=y\frac{\partial{}}{\partial{x}} + x\frac{\partial{}}{\partial{y}} \end{equation}$$ Find it's integral curves and the curve that intersects point $$p = \left(1, 0 \right).$$ Show that $$X(x,y)$$ is tangent to the family of curves: $$x^2...
  36. K

    Integral help (substitution and boundaries)

    Homework Statement Hello, I´ve added an image where the task is shown. I am wondering about the integral bounds here. When doing the substitution, shouldn't the bounds be inf and 0? When using that u = ln x, i.e. u(0) =inf, u(1) = 0. Homework EquationsThe Attempt at a Solution
  37. E

    A Lie derivative of vector field defined through integral curv

    Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted ##\phi _ { t } ( p ) .## Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...
  38. ErikZorkin

    I Counterpart of Feynman integral in de Broglie-Bohm theory

    I am looking for good references / clarifications on the subject. First of all, my question is concerned only with mathematical formulation of something that sort of plays the role of the Feynman path integral of the "standard" QFT. It is not concerned with the physical or philosophical...
  39. stevendaryl

    A Question about equivalence of Path Integral and Schrodinger

    I've seen a proof that the path integral formulation of quantum mechanics is equivalent to solving Schrodinger's equation. However, it appears to me that the proof actually depended on the Hamiltonian having a particular form. I'm wondering how general is the equivalence. Let me sketch a...
  40. S

    I Integral over a region in spacetime

    Hello, can anyone show me if this integral can be evaluated? ##\frac{1}{a_0^2}\int_\Sigma\frac{dy'dz'}{\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg)^2}##
  41. S

    MHB Integral of a portion of spacetime

    Hello, I have difficulty in evaluating this integral. Can anyone assists? $\frac{1}{a_0^2}\int_\Sigma\frac{dy'dz'}{\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg)^2}$
  42. T

    Converting Cartesian to Polar (Double Integral)

    Homework Statement Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy. I am having difficulty finding the bounds for theta and r. Homework Equations I understand that somewhere here, I should be changing to x = r cost y = r sin t I understand that I can solve...
  43. YoungPhysicist

    I Integral power rule explanation

    I am new to the world of calculus and the first thing that I learned is how to calculate the area under the range of a polynomial function, like: $$\int_1^3 x^2 \,dx$$ when I take the intergal of ##x^2##, I get ##\frac{x^3}{3}##due to the power rule, but it doesn’t make sense to me,why would...
  44. E

    Can this integral be simplified?

    Homework Statement ∫∞-∞|∂Ψ/∂x|2dx Homework Equations Can this be simplified at all The Attempt at a Solution
  45. B

    I The integral form of Gauss' theorem

    In many texts I have seen, Gauss theorem has the form of$$\frac{q}{\epsilon_0}=\oint\vec{E}d\vec{A}$$ Why a line integral symbol was used for this surface integral everywhere? The more I see it the more I believe there is something wrong with my understanding about this. I didn't think too much...
  46. M

    I Riemann Integral: Does g(x)=f(x) Almost Everywhere?

    Let f(x) be a bounded continuous function on [0,1]. Let g(x)=f(x) on all rational points in [0,1]. Let g(x) be Riemann integrable on [0,1]. Does g(x)=f(x) almost everywhere in the interval? If so - proof? If not -counterexample.
  47. V

    MHB Find Analytic Expression for Integral with Approximations

    Find the closed form (or) analytic expression form for the following integral $$ \hspace{0.3cm} \large {\int_{0} ^{\infty} \frac{\frac{1}{x^4} \hspace{0.1cm} e^{- \frac{r}{x^2}}\hspace{0.1cm}e^{- \frac{r}{z^2}} }{ \frac{1}{x^2} \hspace{0.1cm} e^{- \frac{r}{x^2}}+ \frac{1}{y^2}...
  48. jonathanm111

    Vector Calculus, setting up surface area integral.

    The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49 this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.
  49. J

    Definite Integral of Product/Composite Function Given Graph

    Homework Statement Given the graph of f(x) shown below, find the value of the integral. Photo attached. Homework Equations [/B] ∫23 5x·f(x2)dx The Attempt at a Solution [/B] I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x2))...
  50. Lapse

    Integral for Current through an Inductor -- Not Working

    Homework Statement [/B] Calculate, and plot along with (on the same plot) the voltage seen below, the current flowing in the following circuit using the integral relationship between the voltage across an inductor and the current through the inductor. Verify your hand calculations and plot...
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