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. 1

    Some integrals I just don't know how to do

    Homework Statement Knowing what I do (U-Substitution, beginning Integration by Parts) what would you do for these? (ln t)^2 (sin t)^2 Homework Equations The Attempt at a Solution All I have been able to do is change these to (ln t)(ln t) and then try by parts, but I just end...
  2. M

    A Question about Defining Logarithms as integrals

    When Logarithm are Defined as integral and the Exponetial functions are defined to be its inverses , then What can prove ? or why ? a^n = \underbrace{a.a.a...a}_{n-times} : n\in N also Why we define rational exponents as roots? I am sorry If my Question is silly. IS my Question...
  3. Z

    Numerical double integrals along discontinuous surfaces

    I posted this in the aerospace engineering forum but I think it may get more replies here: I've been trying to compute the bending-torsion coupling constants for a wing, B1, B2 and B3. The expression for this is \begin{bmatrix} B_1 \\ B_2 \\ B_3 \end{bmatrix} = \iint (y^2 +...
  4. Z

    Evaluating double integrals for bending-torsion coupling constants

    I've been trying to compute the bending-torsion coupling constants for a wing, B1, B2 and B3. The expression for this is \begin{bmatrix} B_1 \\ B_2 \\ B_3 \end{bmatrix} = \iint (y^2 + z^2)\begin{bmatrix} y^2 + z^2 \\ z \\ y \end{bmatrix}dydyz where x is in along the wingspan direction, y...
  5. D

    Can you clarify this step (charge density and integrals).

    Hi, if you have the book: Physics for Scientists and Engineers 8E, Serway Jewett On page 675 (Chapter 23), Example 23.8 there is a step taken during the integration I don't understand: How do you go from "2r dr" in the numerator to d(r^2)? If there is info or link to the property of...
  6. J

    How shall we call these types of integrals in Complex Analysis?

    \mathop\int\limits_{\infty} \log[(z-1)(z+1)]dz=A(z)\biggr|_0^0=4\pi i The infinity symbol below the integral is a positive-oriented, closed, and differentiable path over the function looping around both branch-points and A(z) is the antiderivative of the integrand. I mean would that hold for...
  7. E

    Feynman's book: Quantum Mechanics and Path Integrals

    Hello, I tried to read Feynman's book: Quantum Mechanics and Path Integrals but it is so difficult. Is it a really important book if you want to learn Quantum Mechanics? If so what should I do in preparation to read it? Thanks
  8. Demystifier

    Path integrals and foundations of quantum mechanics

    It is frequently stated that path integral formulation of quantum mechanics is equivalent to the more traditional canonical quantization. However, I don't think it is really true. I claim that, unlike canonical quantization, path integral quantization is not self-sufficient. That's because...
  9. J

    New Calculus student needs help with integrals

    I'm a new Calculus student, and it's killing me! Please help! :( Thank you Homework Statement Find to 2-decimal place of the following integrals. Homework Equations 1. (Upper limit= 33, lower limit= 18) 1 / ([7x+8]^0.25) dx 2. (Upper limit = 33, lower limit = 5) (x^1.6 + 26)...
  10. M

    Use the properties of integrals to verify the inequality

    Homework Statement ∫(from pi/4 to pi/2)sin x/x ≤ 1/√2. Homework Equations The Attempt at a Solution I know the pi/4≤x≤pi/2 and so 1/√2 ≤ sin x ≤ 1 and i have tried to manipulate this to no end and it has annoyed the living daylights out of me
  11. W

    Mathematica Mathematica GPU quasi monte carlo integrals using CUDA

    hello all! I just got a new computer with an Nvidia card, and am now able to do some GPU parallel processing inside mathematica using CUDA. My main interest is in taking tons of moderate accuracy (3-4 digits) numerical integrals. I've been using QMC in MMA and that's been working well...
  12. Y

    Line integrals distance elements

    in line integrals we always need a vector element of distance. I can't understand the difference between ds and dr. is ds for all kinds of paths (even curly ones) and dr only for straight lines, or theyre the same? I am confused, or maybe dr is just the magnitude of ds, and the vector here is...
  13. K

    Changing the limits on Integrals

    I'm confused as to when to change the limits on a definite integral. Ex. Integral with the limits a=1, b=5, 3/(x+1)dx I set u = x+1 and du = dx I used u-substitution and everything worked out fine. However for this one... Ex. Integral with the limits a = 0, b = 2...
  14. G

    Calculus II improper integrals

    Hi, I was wondering if it was really necessary to evaluate improper integrals with limits? Could anyone really say I was wrong if I did something like find the area bounded by the region y=1/x^2, x=2, and the x-axis integral[1,inf] dx/x^2 = (-1/x)|[2,inf] = (-0)-(-1/2)=1/2 Like I don't...
  15. N

    Properties of Summations and Integrals question

    Let's say we have the statement \sum^{\infty}_{0}f(x)=\frac{\sum^{\infty}_{0}g(x)}{\sum^{\infty}_{0}h(x)} does this imply that \int^{\infty}_{0}f(x)=\frac{\int^{\infty}_{0}g(x)}{\int^{\infty}_{0}h(x)}? Also if \sum^{\infty}_{0}f(x)=\sum^{\infty}_{0}g(x) does this imply that f(x)=g(x), or...
  16. M

    Questions about double and triple integrals

    Hey, I was just going through my vector calc textbook for this year and everything was going well until I reached double and triple integrals. My problem is the whole symmetry thing; when does (forgive me, I can't figure out the symbols) the integral from a to b become twice the integral from...
  17. F

    Wow I am stumped What is the difference between these two integrals?

    Homework Statement Suppose s'(t) is a velocity function, then which of the integral will give you the total distance? (1) \int_{a}^{b} \sqrt{1 + [s'(t)]^2} dt (2) \int_{a}^{b} |s(t)| dt The Attempt at a Solution No clue at all... the first is arc length, so it is like...
  18. D

    Vector Calc: Find the volume [using triple integrals]

    1. Find the volume, using triple integrals, of the region in the first octant beneath the plane 2x+3y+2z = 6 2. http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx SOLUTION: 1. Assume X and Y are 0. Solve for Z: 2(0)+3(0)+2z=6 => z=3 (0,0,3) 2. Assume X...
  19. S

    How to calculate elliptic integrals in MATHCAD?

    Hi, I wanted to calculate elliptic integrals (K & E) for a given function in Mathcad. I was not able to find the appropriate function. I am using Mathcad 15. Regards, -sgsawant
  20. N

    Change of Variables in Multiple Integrals

    The problem is: R is the parallelogram bounded by the lines x+y=2, x+y=4, 2x-y=1, and 2x-y=4. Use the transformation u=x+y and v=2x-y to find the area of R. I am not sure how to complete this problem. My first issue is that I don't know how to convert the transformation functions into...
  21. S

    Solving integrals with absolute values

    Homework Statement solve the integral [abs(x+1)(3+abs(x))]/(x+1) between -3 and 1 Homework Equations The Attempt at a Solution when x<-1 then [abs(x+1)(3+abs(x))]/(x+1) = [-(x+1)(3-x)]/(x+1) = -(3-x) when -1<x<0 then [abs(x+1)(3+abs(x))]/(x+1) = (x+1)(3-x)/(x+1) = 3-x when x>0...
  22. G

    Calculus II - Trigonometric Integrals - Evaluate Integral tan(x)^5*sec(x)^4 dx

    Homework Statement Hi, I'm trying to solve this problem and guess I'm doing something wrong. Evaluate Integral tan(x)^5*sec(x)^4 dx Homework Equations integral tan(x) dx = ln(|sec(x)|) integral tan(x)^n dx = tan(x)^(n-1)/(n-1) - integral tan(x)^(n-1) dx tan(x)^2+1=sec(x)^2 The Attempt at...
  23. G

    Evaluating Integrals Using Trigonometric Function Substitutions Question

    Hi, I just had this idea pop into my head... Can you use a trig sub with a reference triangle who has sides equal to zero? or more like a value close to zero such as dx or da or something? For example integral 1/sqrt(9+dx^2) (dx)^2 would have a reference triangle were the hypotenuse is...
  24. G

    Calculus II - Trigonometric Integrals

    Homework Statement Evaluate integral csc(x)^4/cot(x)^2 dx Homework Equations The Attempt at a Solution Apparently I'm doing something wrong, what I'm not sure, thanks for any help My Answer: 2*tan(x) - (sec(x)^2*tan(x))/3 + c integral csc(x)^4/cot(x)^2 dx used fact that...
  25. G

    Calculus II - Trigonometric Integrals HARD

    Homework Statement Evaluate integral sin^(-3/2)(x)cos^3(x) dx Homework Equations tan(x)=sin(x)/cos(x) sin^2(x)+cos^2(x)=1 sin^2(x)=(1-cos(x))/2 cos^2(x)=(1+cos(2x))/2 integral cos(x)dx = sin(x) + c integral sin(x)dx = -cos(x) + c d/dx sin(x) = cos(x) d/dx cos(x) = -sin(x) a^m/a^n=a^(m-n)...
  26. L

    Question about complex integrals

    Homework Statement Hey people got a question here about complex integration, not really sure how to do it so hope someone out there could help me! Evaluate the complex integrals ∫ c { (zbar)^2 +1 } dz...and...∫ c { zcos(z^2) - ie^2z } where c is the contour joining 0 to 2i along...
  27. G

    Calculus II - Trigonometric Integrals

    Homework Statement Evaluate integral( sin^3(x) cos^5(x) ) dxHomework Equations sin^2(x) + cos^2(x) = 1 integral x^n dx = x^(n+1)/(n+1) + c d/dx cos(x) = -sin(x) a^n*a^m=a^(n+m) The Attempt at a Solution I got -cos^6(x)/6+cos^8(x)/8+c Apparently I did something wrong SEE ATTACHMENT Thank...
  28. G

    Calculus II - Trigonometric Integrals

    Homework Statement Apparently I'm doing something wrong. I'm kind of lost as to what because I looked over my work several times. Homework Equations sin^2 x = ( 1 - cos 2x )/2 cos^2 x = ( 1 - sin 2x )/2 integral sin(x) dx = -cos(x) integral cos(x) dx = sin(x) The Attempt at a...
  29. S

    Laplace Transforms for improper integrals?

    Homework Statement I (came up with)/(heard about) a way of using Laplace transforms that I didn't think about before. The problem is that it doesn't work for some reason. Look at following integral: I = \int_{0}^{\infty }sin(t)dt Say that you had no idea how to integrate something...
  30. W

    Absolute Value Integrals

    Homework Statement ∫ |x^2 -9| [0-4] Homework Equations The book answer states the same EXCEPT splits into [0-3] and [3-4]. Other problems split the integral perfectly in half for absolute values...why would it differ and are there rules to figure this out? Larson's Calculus has no...
  31. 1

    Is It Necessary to Substitute u in Integrals?

    I received no credit, resulting in an 84 for a few integral problems. I had correct final answers for everything. When I confronted my professor about this, he said it was because I didn't actually put "u" and "du" into the integral. Is that really always necessary? Why actually put the u in...
  32. P

    Mathematica Mathematica®: performing a varying number of multiple integrals

    Hello everyone. In Mathematica® I want to numerically integrate a function of k variables (k varies) with respect to all of them. Does anyone of you know a way to do that? I tried the following simplified example. k = 5; int[x_] := x[[1]] + x[[2]] + x[[3]] + x[[4]] + x[[5]] ; (* My...
  33. T

    Calculating flux using surface integrals.

    This isn't homework. I've been restudying vector calculus from the beginning to end on my free time and got stuck on this problem. I am not sure what I'm doing wrong, but it may be a calculation error since it has so much calculation involved. Homework Statement Evaluate the surface integral...
  34. Rasalhague

    Substitution of variables in improper integrals

    What principles apply when making a substitution of variables in an improper integral. I gather that a substitution of variables can change an impoper integral to a proper integral. Can substitution also change a proper integral into an improper integral? Suppose I know that a pair of integrals...
  35. A

    Change of varibles in integrals (More than 1 question)

    Homework Statement Homework Equations The Attempt at a Solution Why I can't integrate\theta from 0 to 2\pi? Then integrate \varphi from 0 to \pi. It seems it can also generate a sphere.
  36. A

    Integrating Trigonometric Functions: How to Remember and Use Identities?

    Homework Statement I have a problem in part b Homework Equations The Attempt at a Solution How to integral the function?
  37. M

    Why are my integrals giving different results for the same function?

    I've been solving this exercise and I came to a point when one function can get two different integrals: Am I doing something wrong? Because both functions are the same, and the integrals (indefinite) are really different. This is a huge problem, because this is almost the final step of an...
  38. P

    Evaluating Integral with $\theta(t)$ - Sin | Cos

    Suppose I know the value of an integral: \int_0^T cos(\theta)dt = x Is there any way to evaluate the integral \int_0^T sin(\theta)dt solely from this information? EDIT: \theta=\theta(t), i.e. \theta is a function of t.
  39. S

    Kinematics of surface integrals question

    This is a continuum mechanics/fluid dynamics question concerning the time rate of change of a surface integral of a vector field, where the surface is flowing along in a velocity field (like in a fluid). (Gauss's law is for fixed surfaces.) This integral goes by various names in different...
  40. 1

    Indefinite Integrals - which method is preferred?

    Homework Statement \int (x+1)^2 dx Homework Equations The Attempt at a Solution I am just getting into this, and this is a simple problem, but my book and I took two separate routes. My question, essentially, is does any constant you get just "combine" with the "any constant" C...
  41. N

    What is the range for x in a triple integral for a wedge in the first octant?

    Write a triple integral to represent the volume of the solid The wedge in the first octant and from the cylinder y^2 + z^2 <= 1 by the planes y=x, x=0, z=0 First.. i find the range for z..; 0 <=z<= sqrt(1- y^2) then... i find the range for y..; let z =0 0<=y<=1 next, if i...
  42. M

    When will I learn about elliptical integrals

    What class will I start to study these?
  43. QuarkCharmer

    Integrals (u-substitution)

    Homework Statement Homework Equations The Attempt at a Solution I don't understand what exactly is going on here. They let u=(1+x^{2}), so that leaves them with this: \int \frac{x}{u^{2}}dx The derivative of (1+x^{2}) is simply 2x. And so: \frac{du}{dx} = 2x \rightarrow du =...
  44. N

    Double Integrals (polar coordinate)

    Hey there.. i try to solve the question below.. but.. i still didn't get the answer given by my lecturer.. the answer should be.. pi/4(e - 1) where did i do wrong? http://imageshack.us/photo/my-images/215/06072011697.jpg/ http://imageshack.us/photo/my-images/17/06072011699.jpg/...
  45. J

    About bessel function integrals

    hello,everyone i want to know how to solve this bessel function integrals: \int_{0}^{R} J_m-1(ax)*J_m+1 (ax)*x dx where J_m-1 and J_m+1 is the Bessel function of first kind, and a is a constant. thanks.
  46. T

    Gamma and Beta Integrals

    Homework Statement The Gamma and Beta integrals are defined respectively as \Gamma (z) = \int^{\infty}_0 t^{z-1} e^{-t}\;dt B(p,q) = \int^1_0 t^{p-1} (1-t)^{q-1}\;dt. Determine for what values of the complex parameters z, p, q the integrals converge absolutely and explain why. The...
  47. C

    Double Integrals Limits of Integration

    I am just starting to do double integrals and came acorss an issue. I remembered from single integrals when we integrate from limits for say -1 to 1, we can double it and change integration limits to 0 to 1. Now, when is this the case? Basically, when can we not do this?
  48. M

    How to Solve Complex Integrals: Understanding the Process?

    I'm integrating 1/(z-1/2) over the closed disk w/ radius = 3 centered at 0. I've seen other problems where the final answer was i2pi times f(w) - here w =1/2. Since f(z) is equal to 1. Is the final answer just i2pi? Next up: I have the integral of dt/(2 + sint) the problem then tells me...
  49. S

    Self studying little Spivak's, stuck on Schwartz ineq. for integrals

    Homework Statement In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight. Problem 1-6...
  50. D

    Finding integrals of the product of trig functions

    Homework Statement I've come across integrals of exponential and trig functions and I have no idea how to do them. Integration by parts doesn't really work because they just derive into either e or another trig function. One of them is \intsin(a)*sin(b - a)da Another is \inte(a)*sin(a)da...
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