What is Integrals: 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. R

    Multiple integrals in polar form

    Homework Statement do you see how the integral of r is .5? I don't get how that follows?
  2. R

    Solving Multiple Integrals - Understanding Triangle Area Calculation

    Homework Statement The Attempt at a Solution I understand the steps, although it took quite a while, but what I don't understand is that a triangle with base 2 and height 2, it's area is 2. With two triangles of that size the area should be 4. The books says the area is 8.
  3. N

    Surface terms in loop integrals (2D)

    Hi everyone, Say I have a 2D one loop integral of the form $$ I^s_n(\Delta)=\int d^2 l \frac{(l^2)^s}{(l^2-\Delta)^n} $$ Using that ##1 = \frac{1}{2} \partial_\mu l^\mu##, I can relate say $$I^1_2(\Delta)=I^0_1(\Delta)$$ + total derivative term. In dimensional regularization one usually...
  4. M

    Does the Definite Integral Equal Zero for a Continuous Function?

    Let f : R to R be a continuous function, and suppose that definite integral from m to n |∫(m to n)f(x)dx|≤(n-m)^2 for every closed bounded interval [m, n] in R. Then is it the case that f(x) = 0 for all x in R? I tried using fundamental theorem of calculus but got stuck, since I only got that...
  5. N

    Integrals involving Secant & Tangent Derivation

    Homework Statement If the power of the secand is even and positive.. \int sec^{2k} x tan^{n} x dx = \int (sec^2 x)^{k-1} tan ^n x sec^2 x dx The Attempt at a Solution The way I see it, sec^{2k} x = sec^2 x dx * sec^k x dx the next step seems to be to break down sec^k, but on closer...
  6. T

    Indefinite integrals. Arriving at different results.

    *SOLVED* Indefinite integrals. Arriving at different results. Mistake found. Thanks!, everything looks correct now.
  7. H

    Confusion on sign convention for surface integrals

    Homework Statement Compute the surface integral for F = [3x^2, y^22, 0] and S being a portion of the plane r(u,v)=[u,v,2u+3v], 0≤u≤2, −1≤v≤1.The Attempt at a Solution I managed to get the correct answer, because with some luck I defined the normal in the correct direction. I am just confused...
  8. C

    Surface integrals of vector fields

    The integral for calculating the flux of a vector field through a surface S with parametrization r(u,v) can be written as: \int\int_{D}F\bullet(r_{u}\times r_{v})dA But what's to stop one from multiplying the normal vector r_{u}\times r_{v} by a scalar, which would result in a different...
  9. GreenGoblin

    MHB Choosing Limits for Volume Integrals

    Help choose the limits of the following volume integrals: 1) V is the region bounded by the planes x=0,y=0,z=2 and the surface z=x^2 + y^2 lying the positive quadrant. I need the limits in terms of x first, then y then z AND z first, then y and then x. And also polar coordinates, x=rcost...
  10. N

    Double integrals - Change of variables

    Homework Statement Find the area in the positive quadrant of the x-y plane bounded by the curves {x}^{2}+2\,{y}^{2}=1, {x}^{2}+2\,{y}^{2}=4, y=2\,x, y=5\,x The Attempt at a Solution This is a graph of the region: http://img21.imageshack.us/img21/2947/59763898.jpg One thing I was...
  11. B

    Calculate the following contour integrals sing suitable parametr

    Calculate the following contour integrals using suitable parameterisations Homework Statement 1)##\oint \frac{1}{z-z_0} dz## where C is the circle ##z_0## and radius r>0 oriented CCW and ##k\ge0## 2) ##\int_c |z|^2 dz## where C is the straight line from 1+i to -1 3. Relevant equations...
  12. G

    Contour Integrals: Calculate 0 to 1+i

    Homework Statement Calculate the following contour integrals \int_{c1} (x^3-3xy^2 ) + i (3yx^2 - y^3) where c1 is th line from 0 to 1+i Homework Equations The Attempt at a Solution a earlier part of the question asked if it was analytic. using Cauchy-Reimann equations i have...
  13. C

    2 exercises on changing variables of double integrals

    Homework Statement a) \int\int_{B}\frac{\sqrt[3]{y-x}}{1+y+x} dxdy, where B is the triangle with vertices (0, 0), (1, 0), (0, 1). b)\int\int_{B}x dxdy where B is the set, in the xy plane, limited by the cardioid ρ=1-cos(θ) The Attempt at a Solution a) Let ψ: \left\{u = y-x, v =...
  14. L

    Weinberg QFT - Inner product relations, Standard momentum, Invariant integrals

    Weinberg in his 1st book on QFT writes in the paragraph containing 2.5.12 that we may choose the states with standard momentum to be orthonormal. Isn't that just true because the states with any momentum are chosen to be orthonormal by the usual orthonormalization process of quantum mechanics...
  15. N

    Calculating regions of double integrals

    Homework Statement Evaluate the follow by first changing the order of integration \int_{x=-1}^{1}\int_{y=x^2}^{2-x^2}dydxThe Attempt at a Solution This is the region we're concerned with: http://www.wolframalpha.com/input/?i=plot%28y%3Dx^2%2C+y+%3D+2+-+x^2%2C+x%3D+1%2C+x%3D+-1%29 The new...
  16. T

    Double Integrals: Changing the order of integration

    Hi, I am able to manipulate and use double integrals, but I am having a bit of mental block when trying to visual how they actually work. First, would you agree that a double integral is simply summing a function over a region by taking lots of tiny squares (or rectangles?) of sides dx...
  17. G

    Evaluating Line Integrals with Green's Theorem

    Homework Statement Let C be the boundary of the region bounded by the curves y=x^{2} and y=x. Assuming C is oriented counter clockwise, Use green's theorem to evaluate the following line integrals (a) \oint(6xy-y^2)dx and (b) \oint(6xy-y^2)dyHomework Equations The Attempt at a Solution...
  18. F

    Evaluating Integrals: Need Help Factorising Denominator

    Homework Statement Evaluate the integral Homework Equations I can substitute and thus end up with The Attempt at a Solution I then expand the denominator out and end up with 1/ However I then assume I need to factorise the top line of that fraction as this will be the...
  19. C

    Evaluating Double Integrals of Odd and Even Functions on a Disk

    Homework Statement Suppose f : ℝ→ℝ and g : ℝ →ℝ are continuous. Suppose that f is odd and g is even. Define h(x,y) : f(x)*g(y). Let D be a disk centered at the origin in the plane. What is ∫∫h(x,y)dA? D The Attempt at a Solution I know there's probably a trick to it. Is it 0...
  20. R

    Iterated Integrals Question, using Tonelli and/or Fubini

    Homework Statement Let J0(x)=2/\pi\int0\pi/2cos(xcos[y])dy. Show that \int0∞J0(x)e-axdx=\frac{1}{sqrt(1 +a^2)}. Homework Equations Tonelli and Fubini's theorems The Attempt at a Solution Basically I'm finding this problem really hard because I've had to teach myself iterated...
  21. K

    Question about orientation and surface integrals

    Homework Statement I'm a bit confused as to how to determine which component must be positive or negative if the question gives you a surface and says the normal vector is pointing outward or inward. Some surfaces have it so that the z component is positive if n is pointing outward and...
  22. M

    Calculating Integrals with Gauss's Theorem

    \oint_S \vec{A}\cdot d\vec{S}=\int_V div\vec{A}dv Suppose region where \vec{A}(\vec{r}) is diferentiable everywhere except in region which is given in the picture. Around this region is surface S'. In this case Gauss theorem leads us to \int_S \vec{A}\cdot d\vec{S}+\int_S \vec{A}\cdot...
  23. N

    Methodology for evaluating Contour Integrals

    Homework Statement I'm a bit uncertain as to how to do these types of integrals. Let γ be any contour from 1 - i to 1 + i. Evaluate the following: ∫ 4z^3 dzThe Attempt at a Solution I did this in three different methods, two of them gave the correct answer, although this could just be a...
  24. H

    How to Evaluate Difficult Double Integrals with Limits in the Range of 0 to 1?

    Evaluate ∫∫xexp(xy)dA, and R (over which the integrand is to be integrated) is {(x,y)|0≤x,y≤1}. Could someone explain how this is to be done.
  25. B

    Holder's inequality for integrals

    Does anyone know a simple proof for holder's inequality? I would be more interested in seeing the case of |∫fg|≤ sqrt(∫f^2)*sqrt(∫g^2)
  26. M

    Improper Integrals, Specifically integration part

    Homework Statement Find the antiderivative of (x*arctan(x))/(1+x^2)^2) The Attempt at a Solution I've had a few attempt at this (I've been working on it an embarrassingly long time) but i felt most on track doing it by parts. Here's how i went u = arctan(x) du = 1/(1+x^2)*dx...
  27. 1

    How bad is this statement regarding the Fundamental Theorem for Line Integrals?

    State the Fundamental Theorem: Let F be a vector field. If there exists a function f such that F = grad f, then \int_{C} F \cdot dr = f(Q) - f(P) where P and Q are endpoints of curve C. _________________________________ I didn't receive any credit for this answer. Admittedly...
  28. H

    Convergence of improper integrals with parameters

    I'm having a lot of trouble with the subject. Here's one example I'd like explained. F(t_1, t_2) = \int \limits_0^1 x^{t_1}\ln^{t_2}\frac{1}{x} dx The book asks to find for what \vec{t} F converges. The answer is \vec{t}\in(-1; \infty)^2, but I don't see how to get that. In general, what...
  29. A

    Fundamental Theorem for Line Integrals

    Vector field F(bar)= <6x+2y,2x+5y> fx(x,y)= 6x+2y fy(x,y)= 2x+5y f(x,y)= 3x^2+2xy+g(y) fy(x,y)=2x+g'(y) 2x+g'(y)= 2x+5y g'(y)= 5y g(y)= 5/2*y^2 f(x,y)=3x^2+2xy+(5/2)y^2 Then find the \int F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1 I'm stuck on finding the last part for the F(bar)...
  30. G

    Riemann integrals and step functions

    Prove the following: If f is Riemann integrable on an interval [a,b], show that ∀ε>0, there are a pair of step functions L(x)≤f(x)≤U(x) s.t. ∫_a^b▒(U(x)-L(x))dx<ε My proof: Since f is Riemann integrable on [a,b] then, by Theorem 8.16, ∀ε>0, there is at least one partition π of the interval...
  31. S

    Using Cauchy Schwartz Inequality (for Integrals)

    Homework Statement Suppose \int_{-\infty}^{\infty}t|f(t)|dt < K Using Cauchy-Schwartz Inequality, show that \int_{a}^{b} \leq K^{2}(log(b)-log(a)) Homework Equations Cauchy Schwartz: |(a,b)| \leq ||a|| \cdot ||b|| The Attempt at a Solution Taking CS on L^{2} gives us...
  32. A

    How can I convert discrete sums to integrals using spline interpolation?

    So kind of like this thread, I'm looking to convert a discrete sum to an integral. My idea thus far has been to arrive at a function via spline interpolation. I'm doing a few different types of sums, but the first ones look like \displaystyle a=\sum_{i=1}^{100}{data[1]*data[4]} where data...
  33. M

    Find the Volume of water in pool using double integrals

    Homework Statement The depth of water in a swimming pool fits the equation f(x,y) = 2sin (x/20 - 7) - 3 cos ( x-3 /5)+8 when 0<=x<=20 and the sides of the pool fir the equations y(x) = 10-(x-10)^2/10 and y(6)= (x-10)^2/20 -5 Find the volume of the water in the pool using a double integral...
  34. T

    Finding Limits for Triple Integrals: How to Solve for the Intersection of Planes

    Homework Statement Use a triple integral to find the volume of the region. Below x+2y+2z=4, above z=2x, in the first octant. Homework Equations V=∫∫∫dV=∫∫∫dxdydz The Attempt at a Solution I have no clue where to begin as to finding those darn limits to integrate with. I'm sure...
  35. C

    Calculating indefinite integrals

    Homework Statement a) S 13(4^x + 3^x)dx b) S (cosx + sec^2x)dx c) S (3-(1/x))dx d) S e^(7x)dx Homework Equations The S is supposed to be the integration sign The Attempt at a Solution Are these correct or at least close? a) = 13((4^x)/(ln(4) + (3^x)/(ln(3))) + C b) =...
  36. F

    Polar Coordinates to evaluate integrals

    Homework Statement Use Polar coordinates to evaluate were C denotes the unit circle about a fixed point Z0 in the complex plane The Attempt at a Solution I've only used polar integrals to convert an integral in sin and cos into one in therms of z, find the residues and then use the...
  37. I

    Some questions concerning asymptotic expansions of integrals

    I've started self-teaching asymptotic methods, and I have some theoretic questions (and lots of doubts!). 1. Say I have the asymptotic expansion f(x) \asymp \alpha \sum_n a_n x^{-n} for x large, where \alpha is some prefactor. How can I estimate the value of n for the term of...
  38. W

    Applications of Line integrals

    other than the physics (work) what are the applications of line integral? particularly does it have any use in finance or economics?
  39. D

    Understanding the meaning of multiple integrals

    Homework Statement I am currently taking calc III and we have starting getting into double and triple integrals. I was wondering what you are actually doing when you take a double or triple integral? And what the difference is. I understand that you find area with a single integral and find...
  40. 1

    I thought I understood double integrals until I saw this

    Homework Statement http://img710.imageshack.us/img710/4764/doubleintegral.png Homework Equations The Attempt at a Solution Now, my understanding of the region is that x spans from the line x = y to x = 1, and that given that parameter, the applicable y's are 0 to 1. In other...
  41. 1

    How to Calculate the Mass of a Pyramid Using Triple Integrals?

    Homework Statement Find the mass m of the pyramid with base in the plane z = 9 and sides formed by the three planes y = 0 and y - x = 5 and 6x + y + z = 28, if the density of the solid is given by δ(x,y,z) = y. Homework Equations The Attempt at a Solution This problem is driving...
  42. V

    Real integrals with complex coefficients

    I'm curious about the validity of various techniques from good old calculus in one real variable when dealing with complex coefficients. I know enough complex analysis to know that the rules change when dealing with complex variables, but I'm curious about the case when the variables are still...
  43. S

    Complex Integrals - Poles of Integration Outside the Curve

    Homework Statement \int_{|z-2i|=2} = \frac{dz}{z^2-9} 2. The attempt at a solution I know that the contour described by |z-2i|=2 is a circle with a center of (0,2) (on the complex plane) with a radius of 2. The singularities of the integral fall outside of the contour (z+3 and...
  44. M

    Polar coordinates and multivariable integrals.

    Homework Statement Im righting this down for my roommates since he's having tons of trouble trying to figure this out and I can't answer it. also sorry for having to hotlink it. http://i.imgur.com/afShz.jpg the equation is on the image since its very difficult to type it all out...
  45. M

    Interpretation of dx as the differential of x for Indefinite Integrals

    Interpretation of "dx" as the differential of x for Indefinite Integrals This question is concept-as-opposed-to-calculation based. I understand that when one sees the integral sign, followed by f(x)dx, that we can think of this as the indefinite integral, or antiderivative of f(x), with...
  46. X

    Contour Integrals, which contour to use?

    Homework Statement http://img404.imageshack.us/img404/3952/contf.png The Attempt at a Solution Is there a set of rules or postulate that refer to which contour to use for specific integrals? I tried to use the residue theorem for the first integral but I didn't get the right answer
  47. J

    Need help with integrals from Landau-Lifshitz vol. 1 problems

    Hi! I'm new to this forum and forums in general, so please be forgiving. I am currently going through problems in Landau-Lifgarbagez's vol. 1 (Mechanics) and encountered two integrals I can't solve. The physical basis of the problem is crystal clear, but I can't do the final computation. The...
  48. S

    MHB Exploring the Depths of Integrals

    integrals (lots of them!) 1. $\displaystyle \int_{0}^{1}\frac{\sin(\ln x)}{\ln x}dx$ 2. $\displaystyle \int_{0}^{1}\frac{\arctan(x)}{1+x}dx$ 3. $\displaystyle \int_{0}^{\infty}\frac{\ln(1+x^2)}{1+x^2}dx$ 4. $\displaystyle \int_{0}^{\infty}e^{- \left( x^2+\dfrac{1}{x^2}\right)}dx$ 5...
  49. O

    Maxwell's, integrals, current, elements, delta phi and confusion

    I'm working on an online EECS course, and to be frank some of it is going straight over my head - but at the same time parts of it are far below my current knowledge, so I want to work and stick with it. The speaker is working through proving current and voltage - to arrive at Kirchoff's...
  50. J

    Solving integrals with the table of integrals

    Homework Statement ∫e2xarctan(ex)dx Homework Equations From the table of integrals: #92 ∫utan-1udu = (u2+1)/2)tan-1-u/2 + c or #95 ∫untan-1udu = 1/(n+1)[un+1tan-1-∫ (un+1du)/(1+u2) , n≠-1 The Attempt at a Solution The answer is 1/2(e2x+1)arctan(ex) - (1/2)ex + C I don't...
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