Integrals Definition and 1000 Threads

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. F

    Physical motivation for integrals over scalar field?

    I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found: If you want to know the final temperature of an object that travels through a medium described with a temperature field then you'll need a line integral It appears to me that...
  2. A

    Does Gauss' Law use line integrals or surface integrals?

    In my physics textbook, I see Gauss' Law as https://upload.wikimedia.org/math/0/3/5/035b153014908c0431f00b5ddb60c999.png\ointE dA but in other places I see it as...
  3. I

    Evaluating Fresnel Integrals Using Euler Formula

    Homework Statement Evaluate the following integrals C = 0inf∫cos(x2) dx and S = 0inf∫sin(x2) dx Homework Equations [/B] Hint: use Euler formula to write the integral for F = C + iS. Square the integral and evaluate it in polar coordinates. Temporary add a convergence factor. Answer: C = S =...
  4. P

    Surface Integrals: Evaluate A.n dS on 2x+y=6 Plane in 1st Octant

    Homework Statement Evaluate integral A.n dS for A=(y,2x,-z) and S is the surface of the plane 2x+y=6 in the first octant of the plane cut off by z=4 Homework Equations Integral A.n dS The Attempt at a Solution The normal to the plane is (2,1,0) so the unit normal vector is 1/sqrt3 (2,1,0)...
  5. P

    Area of z^2=xy inside Hemisphere: Surface Integrals

    Homework Statement Find the area of the part of z^2=xy that lies inside the hemisphere x^2+y^2+z^2=1, z>0 Homework Equations da= double integral sqrt(1+(dz/dx)^2+(dz/dy)^2))dxdy The Attempt at a Solution (dz/dx)^2=y/2x (dz/dy)^2=x/2y => double integral (x+y)(sqrt(2xy)^-1/5) dxdy Now I'm...
  6. benorin

    Need some kind of convergence theorem for integrals taken over sequences of sets

    I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...
  7. M

    Finding Volumes of Sphere & Circular Cone: Alpha from 0 to Pi

    Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle alpha with its vertex at the origin. Show that in the limits where alpha = 0 and alpha = pi that your expression gives the...
  8. H

    Double Integrals: Interchangeability and Improper Integration Explained

    I started learning double integrals in this semester but I got some problems on it. Is it matter if the order of integration interchange? For example, V=∫f(x,y)dydx, can we rewrite the equation as V=∫f(x,y)dxdy? If not, could you explain on it little bit:sorry: Thanks!
  9. E

    Integrals to Solve Area and Center of Mass of a Cut Circle

    Homework Statement I am after finding the centroid of the remaining area (hatched) when a circle is cut by a line. I made a diagram in CAD that demonstrates the problem. The idea is that, starting from the bottom of the circle, a cut is taken leaving a remaining shape whose area and...
  10. S

    Python Are errors important in calculating integrals in Python?

    I am reading Mark Newman's Computational Physics textbook. He goes over calculating integrals with Simpsons's Rule and the Trapezoidal Rule, and then he goes over calculating their errors. Why would I have to ever worry about the error of the integral? He has the chapters online at his website...
  11. PhysicsBoyMan

    Solving Integrals with Exponents

    Homework Statement Evaluate the integral to find the area. Homework Equations The Attempt at a Solution[/B] gifs upload So I know how to find an anti-derivative for the most part. Here it's tricky because my equation has an exponent AKA square root. I tried to use the chain rule with...
  12. H

    Propagator in the derivation of path integrals

    Why isn't (5.298) the following instead? ##K(x, t_1; x', t_0) = \delta(x-x')\,e^{-\frac{i}{\hbar}H(t_1-t_0)}## My reasoning: Since \Psi(x, t_1) = e^{-\frac{i}{\hbar}H(t_1-t_0)}\,\Psi(x, t_0)\\ = e^{-\frac{i}{\hbar}H(t_1-t_0)}\,\int\delta(x-x')\Psi(x', t_0)\,dx' The exponential operator with...
  13. B

    Delta Function Integration: Justified or Fudging?

    Hello, I feel like I am fudging these integrals a bit and would like some concrete guidance about what's going on. 1. Homework Statement Evaluate ##I = \int_{-1}^{1} dx \delta'(x)e^3{x} ## Homework EquationsThe Attempt at a Solution [/B] I use integration by parts as follows, ##u =...
  14. T

    Surface area of a sphere with calculus and integrals

    Homework Statement How do I find the surface area of a sphere (r=15) with integrals. Homework Equations Surface area for cylinder and sphere A=4*pi*r2. The Attempt at a Solution I draw the graph for y=f(x)=√(152-x2). A circle for for positive y values which I rotate. I will create infinite...
  15. N

    Upper and lower bound Riemann sums

    Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...
  16. M

    Double integrals: cartesian to polar coordinates

    Homework Statement Change the Cartesian integral into an equivalent polar integral and then evaluate. Homework Equations x=rcosθ y=rsinθ I have: ∫∫r2cosθ dr dθ The bounds for theta would be from π/4 to π/2, but what would the bounds for r be? I only need help figuring out the bounds, not...
  17. P

    Finding volume of a modified sphere using integrals

    Hello guys, new member here. I've got a calculus project due Tuesday that I could use some help on. I won't bore you with the all details of the project, but first let's imagine an olive in the shape of a perfect sphere (with a radius always bigger than 6mm) that goes through a set of blades...
  18. F

    MHB Half wave symmetry and integrals

    If I have a periodic wave x(t) with half-wave symmetry, it means that: x(t + T0/2) = -x(t) where T0 is the period of the wave. Would this automatically lead to the conclusion that X(t + T0/2) = -X(t) where X'(t) = x(t), i.e X(t) is the integral of x(t). ?
  19. L

    Riemann's proof of the existence of definite integrals

    Hello, Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...
  20. E

    MHB Solving Double Integrals: Order of Integration Explained

    Hey guys, need your help and hope someone takes the time: I need to solve the double integral by changing the order of Integration. I would really appreciate if you could illustrate the way of how to compute the solution. Best Estelle :)
  21. G

    Proving Asymptotic Comparisons of Integrals

    Homework Statement Let ##f## be piecewise continuous from ##[0,+\infty[## into ##V = \mathbb{R} ## or ##\mathbb{C}##, such that ## f(x) \longrightarrow_{ x\rightarrow +\infty} \ell ##. Show that ## \frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell## Homework...
  22. LAZYANGEL

    Integrals: Fun to Solve Analytically!

    Hey folks, found a couple of interesting integrals and was able to solve one of them ANALYTICALLY! That means no numerical solutions needed. $$\int_{0}^{\frac{\pi}{2}} \frac{1}{1+(tan(x))^{\sqrt{2}}} dx$$ $$\int \frac{1}{1+e^{\frac{1}{x}}} dx$$ The first one I solved and will reveal analytic...
  23. N

    Compute upper and lower integral

    Homework Statement Let f(x) = x^2 and let P = { -5/2, -2, -3/2, -1, -1/2, 0, 1/2 } Then the problem asks me to compute Lf (P) and Uf (P). Lf (P) = Uf (P) = The Attempt at a Solution Please explain how to solve. I thought that L[f] meant to calculate the lower bound with respect to f(x)...
  24. L

    Complex Integrals, Antiderivatives, Logarithms

    I've been teaching myself a little bit of Complex Variables this semester, and I had a question concerning complex integrals. If I understand correctly, then if a function f has an antiderivative F , then the line integral \int_C f(z) dz is path independent and always evaluates to F(z_1)...
  25. C

    IBP Relations in Feynman integrals

    I am wondering if anyone has experience in using IBP( Integration by parts) identities in the evaluation of Feynman diagrams via differential equations? My question is that I can't seem to understand where equation (4.8) on P.8 of this paper: http://arxiv.org/pdf/hep-ph/9912329.pdf comes from...
  26. Conservation

    Volume of a solid bound by four surfaces

    Homework Statement Compute the volume of the solid bounded by the four surfaces x+z=1,x+z=−1,z=1−y2,z=y2−1 Homework Equations Fubini's theorem? The Attempt at a Solution I have tried to visualize this solid and define the limits; when I attempted to integrate by dxdzdy (in that order), I set...
  27. J

    What Happens When Evaluating Improper Integrals with Limit?

    Homework Statement Evaluating the following formula: The Attempt at a Solution Since the integral part is unknown, dividing the case into two: converging and diverging If converging: the overall value will always be 0 If diverging: ...?
  28. Z

    Calculating integrals using residue & cauchy & changing plan

    Homework Statement \int_{0}^{2\pi} \dfrac{d\theta}{3+tan^2\theta} Homework Equations \oint_C f(z) = 2\pi i \cdot R R(z_{0}) = \lim_{z\to z_{0}}(z-z_{0})f(z) The Attempt at a Solution I did a similar example that had the form \int_{0}^{2\pi} \dfrac{d\theta}{5+4cos\theta} where I would change...
  29. M

    Understanding distances, derivatives, and integrals help?

    Homework Statement Two railroad tracks intersect at right angles at station O. At 10AM the train A, moving west with constant speed of 50 km/h, leaves the station O. One hour later train B, moving south with the constant speed of 60 km/h, passes through the station O. Find minimum distance...
  30. W

    Volume Integrals: What Am I Integrating?

    If I want to integrate the volume inside a cylinder ##x^2+y^2 = 4R^2##, and between the plane (I think it's a plane) ##z= \frac{x^2+3y^2}{R}## and the xy plane, then I know how to convert it to cylindrical co-ords, find the limits of integration, and integrate r dr dθ dz. But exactly what am I...
  31. M

    Is there an error in my textbook?

    Homework Statement Homework Equations Doesn't the integral of sec x tan x equal sec x? The Attempt at a Solution
  32. C

    How to find the radius in each of these integrals?

    Homework Statement Given: y = sqrt(x), y=0, x=3 Find the volume of the solid bounded by these functions, revolved around B) y-axis and C) line x=3. (Disk method) Homework Equations Disk method of finding volume using π ∫ r2dy The Attempt at a Solution Ok so, the part of the problem that I...
  33. Korbid

    Virial expansion: Resolving these integrals

    Hi! From this text: http://arxiv.org/pdf/nucl-th/0004061v1.pdf I need to resolve these integrals. 1) Equation (5), \int e^{-\omega_1/T}d{\vec k}_1 =? where \omega_1=\sqrt{m^2+{\vec k}^2_1} What function is K_2(m/T)? 2) Equation (26), \rho_s(T) Thanks!
  34. B

    Integrals log x in terms of dilogarithm

    Hi members,see attached PdF file.Have you any idea to prove thes integrals Thank you
  35. I

    Can Squaring Integrals Simplify Calculus Problems?

    Hello, When I recently was studying for my calculus I's rules of definite integrals, I was wondering if squaring a definite integral would be the same as integrating twice like in the following:( Definite integral of f(x) from a to b)^2 = definite integral ,from a to b, of the definite...
  36. M

    Using area to evaluate integrals

    Homework Statement Use areas to evaluate the integral f(x)=5x+√(25-x2) on the following intervals a) [-5,0] b) [-5,5] Homework Equations ∫f(x) + g(x) = ∫f(x) + ∫g(x) also Area of a circle = pi(r)2 The Attempt at a Solution [/B] My first several attempts have centered around evaluating the...
  37. Daniel Reyes

    Where Can I Find Software for Calculating Tensor Loop Integrals?

    Hi there! I was wondering if anybody knows what package or software CERN of other particle accelerators use to calculate their theoretical predictions. I need specifically tensor loop integrals of up to rank four and of three and four vertices. Thanks!
  38. gulfcoastfella

    Calculus Where can I find an extensive table of integrals?

    I'm reading through an undergrad physics book, and the author says he looked up the answer to the below integral in a table. I've tried to find tables of integrals with this integral included in them, but have failed so far. Can someone direct me to an exhaustive table of integrals and their...
  39. T

    Geometric interpretations of double/triple integrals

    I am currently taking calculus 3 and I am a little confused about the concept of double and triple integrals. Analytically, it's a breeze. I understand how to set limits, do all calculations, etc. What my question is, when I get an answer, what does the answer "mean"? For example, in this...
  40. DameLight

    "Partial Fractions" Decomposition Integrals

    Hello, I was just introduced to this concept and I have solved a few problems, but I haven't come across any with denominators to a raised power yet. ∫ 1 / [(x+7)(x^2+4)] dx I would appreciate any directed help. 1. from the initial state I have broken the fraction into two assuming that...
  41. bhobba

    Comments - Some Useful Integrals In Quantum Field Theory

    bhobba submitted a new PF Insights post Some Useful Integrals In Quantum Field Theory Continue reading the Original PF Insights Post.
  42. andyrk

    Mean Value Theorem for Definite Integrals

    In the MVT for Integrals: ##f(c)(b-a)=\int_a^bf(x)dx##, why does ##f(x)## have to be continuous in ##[a,b]##.
  43. W

    Defining Integrals over R^R, R^N

    Hi, Just curious: how does one define integrals over ##\mathbb R^{\mathbb R}, \mathbb R ^{\mathbb N} ##? I assume this must be a topic in Functional Analysis. I know a bit about abstract Wiener spaces; is there something else? And I assume the objects that are integrated are operators (linear...
  44. W

    Integrals of the Bessel functions of the first kind

    Hi Physics Forums. I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck. f(x,a) = \int_0^\infty\frac{t\cdot...
  45. P

    Definite Integrals: Prerequisites for Property

    I have a few questions about the following property of definite integrals: $$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$ What exactly are the prerequisites for this property? Should ##c## be a member of ##[a,b]##? Should the function ##f## be defined at ##c##?
  46. H

    Confused about perturbation theory with path integrals

    Homework Statement Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you...
  47. P

    What do Constructivists think of Integrals?

    So if we regard something as only being well defined if we can construct it, does this somehow affect what we think about integrals? The way I understand it, there is absolutely nothing in mathematics that tells you how to actually do an integral. Fundamentally, all we can do is cleverly pull...
  48. kostoglotov

    Insight into determinants and certain line integrals

    I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...
  49. L

    Convergence of improper integrals

    What is the difference between \int_{-\infty}^{\infty} \frac{x}{1+x^2}dx and \lim_{R\rightarrow \infty}\int_{-R}^{R} \frac{x}{1+x^2}dx ? And why does the first expression diverge, whilst the second converges and is equal to zero?
  50. E

    If f is even then then left and right integrals are equal

    Homework Statement If ##f## is an even function then $$\int_{-a}^{0} f = \int_{0}^{a} f$$ Homework EquationsThe Attempt at a Solution My attempt was trying to show the upper sum of both integrals were equal. Take a partition of [-a,0] call it ##P_{1}##, and ##P_{2}## for [0.a]. if we can...
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