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. N

    I Triple equation for integral on a graph

    Hi, so I'm trying to find the volume of a shape using integral, I found the equation of one plane in 3D space but the second one is something like that, which I cannot write in integral as a function: ##\frac{2(2x-a)}{a}=-\frac{2(6y-a\sqrt3)}{a\sqrt3}=\frac{2z-a\sqrt3}{a\sqrt3}## In the 3D...
  2. G

    Can a human calculate this without a calculator?

    my notebook says that we can rewrite the integral $$\int {75\sin^3⁡(x) \cos^2⁡(x)dx}$$ as $$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$ however, i have literally no idea how it got to this point, and i unfortunately can't really provide an "attempt at a solution" for this...
  3. Pipsqueakalchemist

    Engineering (material science) Fatigue life prediction using integral

    So for this question, I understand the math but just wanted to be clear on a few things. So I had this question on my midterm but instead of tensile and compressive stresses, it was tensile and tensile stress. I initially thought that the delta sigma in the integral was the maximum stress so in...
  4. Addez123

    Calculate surface integral on sphere

    I'm supposed to do the surface integral on A by using spherical coordinates. $$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$ $$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$ Now I'm trying to do $$\iint A dS = (rsin\theta cos\phi, rsin\theta...
  5. P

    Integral Calculation: Compute l/sqrt(x2+l2)

    I need compute the integral $$(2\pi)^{-3} \int d^3p e^{-l|p|}e^{i \vec{x} \cdot \vec{p}}$$ The problem does not specified the limits of integration The result is $$\frac{1}{\pi^2} \frac{l}{\sqrt{\vec{x}^2+l^2}}$$I saw the references about t-Student and I had not achieved it. I have tried to...
  6. C

    No Limits of Integration for Electric Field Integral?

    For this problem, The solution is, However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q. However, how would you solve this problem with limits of integration? Many thanks!
  7. P

    I Change of Variables in Double Volume Integral

    In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following. $$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$ $$ \bf z = \bf r^{''} - \bf r^{'} $$ $$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z +...
  8. J

    Integral (-∞ to ∞) of Zero and ∞/∞

    Are both integral on picture below equal zero? I think both are zero, area of zero section under function must be zero. If M=∞, b=∞ , what is reslut? Logically ∞/∞ will be 1..but...
  9. WMDhamnekar

    Computing line integral using Stokes' theorem

    ##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]## The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}## ##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}## Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...
  10. H

    Is an operator (integral) Hermitian?

    Knowing that to be Hermitian an operator ##\hat{Q} = \hat{Q}^{\dagger}##. Thus, I'm trying to prove that ##<f|\hat{Q}|g> = <\hat{Q}f|g> ##. However, I don't really know what to do with this expression. ##<f|\hat{Q}g> = \int_{-\infty}^{\infty} [f(x)^* \int_{-\infty}^{\infty} |x> <x| dx f(x)] dx##...
  11. WMDhamnekar

    Evaluate the line integral

    Author's answer: Recognizing that this integral is simply a vector line integral of the vector field ##F=(x^2−y^2)i+(x^2+y^2)j## over the closed, simple curve c given by the edge of the unit square, one sees that ##(x^2−y^2)dx+(x^2+y^2)dy=F\cdot ds## is just a differentiable 1-form. The...
  12. P

    Introduce an integral function in a spreadsheet (Excel MS)

    I found some interesting equations on cosmology and I was wondering how to introduce the integral in an excel sheet: "Paste ( .443s^3+1)^(-1/2) in for the integrand, type in s for the variable and 1 to 2 for the limits. Press submit, then change 2→3→4→5 and repeat." (from the thread...
  13. N

    I Why can't we just integrate a simple function?

    Can anyone explain to me why the second one is the right? (See the attachment)
  14. T

    Integration of acceleration in polar coordinates

    I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image. My attempts are the following, I proceed using 3 "independent" methods just as you...
  15. Euge

    POTW Contour Integral Representation of a Function

    Suppose ##f## is holomorphic in an open neighborhood of the closed unit disk ##\overline{\mathbb{D}} = \{z\in \mathbb{C}\mid |z| \le 1\}##. Derive the integral representation $$f(z) = \frac{1}{2\pi i}\oint_{|w| = 1} \frac{\operatorname{Re}(f(w))}{w}\,\frac{w + z}{w - z}\, dw +...
  16. M

    Gauss' law in line integral, Q=##ϵ_0 ∮E.n dl=-ϵ_0 ∮∂ϕ/∂n dl##

    I know the Gauss law for surface integral to calculate total charge by integrating the normal components of electric field around whole surface . but in above expression charge is calculated using line integration of normal components of electric field along line. i don't understand this...
  17. S

    B Asking about integral notation

    Why should it be ##\int_{a}^{x} f(t)dt## ? Couldn't it be like this: Let F(x) = ##\int f(x)dx## so ##\int_{a}^{x} f(x)dx## = F(x) - F(a) Thanks
  18. H

    Improper integral of a normal function

    I'm trying to solve an improper integral, but I'm not familiar with this kind of integral. ##\int_{-\infty}^{\infty} (xa^3 e^{-x^2} + ab e^{-x^2}) dx## a and b are both constants. From what I found ##\int_{-\infty}^{\infty} d e^{-u^2} dx = \sqrt{\pi}##, where d is a constant and...
  19. P

    Evaluating the Integral of a Vector Field Using Cauchy-Schwarz Inequality

    Here is my attempt (Note: ## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|## ##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ## Here I am going to surround the above expression with another set of...
  20. M

    Integral of f·g ≠ integral f · integral g [True or False]

    My answer is False! I think must stated "in general," in the beginning of the statement. Cause this could be true if f or g = zero. There may be other cases also. Is my answer right?
  21. SchroedingersLion

    I Integral change of variables formula confusion

    Greetings all. I just got confused by the following. Consider volume integral, for simplicity in 1D. $$ V(A) = \int_{A} dz. $$ If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula $$ V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx...
  22. P

    I What is the purpose of the vector α in the rotational work integral?

    This brief worked example from a textbook section on the method of images is confusing me. Specifically I am confused about the vector α in the integral on the last line. When α (or θ) is an angle, I've only ever seen the vector quantity α (or θ) as a polar vector in the plane. But here...
  23. H

    Help with a line integral please

    ∫zds=∫acos(t)*( (acos(2t))^2+(2asin(t))^2+(-asin(t))^2 )^1/2 dt , (0≤t≤pi/2) Simplified : ∫a^2cos(t)*(cos^2(2t)+5sin^2(t) )^1/2 dt , (0≤t≤pi/2) However here i get stuck and i can´t find a way to rewrite it better or to integrate as it is. Can i please get some help in this?
  24. karush

    Solving this integral with u substitution

    Evaluate ##\displaystyle\int_{0}^{3}\frac{x+3}{\sqrt{x^{3}+1}}dx+5## W|A returned 11.7101 ok subst is probably just one way to solve this so ##u=x^{3}+1 \quad du= 3x^2##
  25. P

    Calculating eletric potential using line integral of electric field

    So, I am able to calculate the electric potential in another way but I know that this way is supposed to work as well, but I don't get the correct result. I calculated the electric field at P in the previous exercise and its absolute value is $$ E = \frac {k Q} {D^2-0.25*l^2} $$ This is...
  26. M

    B Neglected terms in integral sum

    Hello. As is known, we can neglect high-order term in expression ##f(x+dx)-f(x)##. For ##y=x^2##: ##dy=2xdx+dx^2##, ##dy=2xdx##. I read that infinitesimals have property: ##dx+dx^2=dx## I tried to neglect high-order terms in integral sum (##dx^2## and ##4dx^2## and so on) and I obtained wrong...
  27. T

    I Is there an integral version of Newton's law of gravity?

    $$F=G\frac{m_1 m_2}{r^2}$$ is presumably for point masses. If the masses weren't a point masses, then wouldn't you need a version of the formula that sums up the gravity for each infinitesimal portion of the masses? And for my money, "summing up" in physics is integrals, right? So would it be...
  28. WMDhamnekar

    Evaluate the surface integral ##\iint\limits_{\sum} f\cdot d\sigma##

    But the answer provided is ##\frac{15}{4} ## How is that? What is wrong in the above computation of answer?
  29. WMDhamnekar

    MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$

    Evaluate the surface integral $\iint\limits_{\sum} f \cdot d\sigma $ where $ f(x,y,z) = x^2\hat{i} + xy\hat{j} + z\hat{k}$ and $\sum$ is the part of the plane 6x +3y +2z =6 with x ≥ 0, y ≥ 0, z ≥ 0 , with the outward unit normal n pointing in the positive z direction. My attempt to answer...
  30. R

    I Inequality with integral and max of derivative

    Hi. I was reading Lighthill, Introduction to Fourier Analysis and Generalised Functions and in page 17 there is an example/proof where I can't make sense of the following step: $$ \left| \int_{-\infty}^{+\infty} f_n(x)(g(x)-g(0)) \, \mathrm{d}x \right| \le \max{ \left| g'(x) \right| }...
  31. WMDhamnekar

    I What is the equation for line integrals involving the gradient of a function?

    I don't have any idea to answer this question. So, any math help will be accepted. I know ##\nabla fg = f\nabla g + g\nabla f \rightarrow (1) ## But I don't understand to how to use (1) here?
  32. WMDhamnekar

    A Solve Line Integral Question | Get Math Help from Physics Forums

    I don't have any idea about how to use the hint given by the author. Author has given the answer to this question i-e F(x,y) = axy + bx + cy +d. I don't understand how did the author compute this answer. Would any member of Physics Forums enlighten me in this regard? Any math help will be...
  33. WMDhamnekar

    MHB How to prove this corollary in Line Integral using Riemann integral

    . Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $ Hint: Recall that $\left\vert\displaystyle\int_a^b g(x)...
  34. J

    I The asymptotic behaviour of Elliptic integral near k=1

    I'm looking at a proof of the asymptotic expression for the Elliptic function of the first kind https://math.stackexchange.com/questions/4064023/on-the-asymptotic-behavior-of-elliptic-integral-near-k-1 and I'm having trouble understanding this step in the proof: $$ \begin{align*} \frac{1}{2}...
  35. chwala

    Find the indefinite integral of the given problem

    Now the steps to solution are clear to me...My interest is on the constant that was factored out i.e ##\frac{2}{\sqrt 3}##... the steps that were followed are; They multiplied each term by ##\dfrac{2}{\sqrt 3}## to realize, ##\dfrac{2}{\sqrt 3}\int \dfrac{dx}{\left[\dfrac{2}{\sqrt...
  36. WMDhamnekar

    Evaluation of integral having trigonometric functions

    R is the triangle which area is enclosed by the line x=2, y=0 and y=x. Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v## ## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
  37. WMDhamnekar

    I Is this integral using change of variables technique correct?

    Is the answer given above correct?
  38. A

    Is the Result of the Path Integral Positive Due to Negative dx?

    In the book it is mentioned that, in path c, the line integral would be: $$\int \vec{F}\cdot \vec{dr} = A \int_{1}^{0}xy dx = A\int_1^0 x dx = -\dfrac{A}{2}$$. but I think that dx is negative in that case, the result would be positive, right?
  39. chwala

    Find the value of the definite integral

    Find question here, My approach, using cosine sum and product concept, we shall have; ##\cos (A+B)-\cos (A-B)=-2\sin A\sin B## ##⇒\cos D-\cos C=-2\sin\dfrac{C+D}{2} \sin\dfrac {C-D}{-2}## ##⇒-3[\cos(A+B)-\cos(A-B)]=6\sin A sinB## We are given ##A=4θ## and ##B=2θ##, therefore, ##⇒-3[\cos...
  40. WMDhamnekar

    MHB Approximating the Double Integral using Monte Carlo Method

    Write a program that uses the Monte Carlo method to approximate the double integral $\displaystyle\iint\limits_R e^{xy}dA$ where $R = [-1,1] \times [0, x^2]$. Show the program output for N = 10, 100, 1000, 10000, 100000 and 1000000 random points. My correct answer: My Java program...
  41. J

    What is wrong with this flux integral?

    I think the issue is how I parameterize my vector field, but not quite sure. In case you were wondering, this is problem # 27, chapter 16.7 of the 8th edition of Stewart. Thanks for any help.
  42. chwala

    Evaluate the definite integral in the given problem

    My interest is on the highlighted part only. Find the problem and solution here. This is clear to me (easy )...i am seeking an alternative way of integrating this...or can we say that integration by parts is the most straightforward way? The key on solving this using integration by parts...
  43. MevsEinstein

    B Is this identity containing the Gaussian Integral of any use?

    I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?
  44. MevsEinstein

    B Is it possible to find the integral of ##f(x)/x^2##?

    I am creating an integration technique and I have only one step left! I need to integrate ##f(x)/x^2## and then I'll be done. So I want to know if integrating this is possible. Wolfram Alpha can't integrate it, but there are problems that it couldn't solve, so I'm not 100% sure that Wolfram...
  45. greg_rack

    Calculating the Line Integral of F over C: Stokes' Theorem and Symmetry

    From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##. Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C. The latter ends up being 0(I calculated it parametrizing...
  46. H

    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...
  47. A

    Finding the area of a double integral using dxdy instead of dydx

    I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle... So, I was curious how one might...
  48. C

    The integral of (sin x + arctan x)/x^2 diverges over (0,∞)

    My attempt: Disprove. Note that ## \int_{0}^{\infty} \frac{-1 - \frac{\pi}{2} }{x^2} d x \leq \int_{0}^{\infty} \frac{\sin x+\arctan x}{x^{2}} d x ## and that ## \int_{0}^{\infty} \frac{-1 - \frac{\pi}{2} }{x^2} d x ## diverges, hence by the integral Direct Comparison Test, ## J ##...
  49. Lorena_Santoro

    MHB How would you approach this integral?

  50. Ssnow

    I Curiosity: there exists the exponential integral?

    Hi, my question regard the possibility to consider a generalization on the product integral (of type II). The product integral is defined in analogy to the definite integral where instead the limit of a sum there is a limit of a product and, instead the multiplication by ##dx## there is the...
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