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.

View More On Wikipedia.org
Recent contents Top users
  1. L

    Volume of an ellipsoid using double integrals

    Homework Statement Using double integrals, calculate the volume of the solid bound by the ellipsoid: x²/a² + y²/b² + z²/c² = 1 2. Relevant data must be done using double integrals The Attempt at a Solution i simply can't find a way to solve this by double integrals, i did with triple...
  2. powerof

    Conquering the Infinity Limit: Integrals and Exponential Functions

    lim_{x\rightarrow + ∞} \frac{\int^{x^3}_{0} e^{t^2}dt}{x \int^{x^2}_{0} e^{t^2}dt} Attempt at a solution: I don't really know where to start. Any hints?
  3. I

    Double Integrals with Polar Coordinates

    Homework Statement Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 47 - 5x2 - 5y2 and the plane z = 2. Homework Equations x2 + y2 = r2 x = rcosθ y = rsinθ The Attempt at a Solution I substituted the z = 2 into the equation given, 2 = 47 -...
  4. D

    Pattern recognition on integrals questions

    Hello users, I would like to know when do you use pattern recognition over integrals Someone told me it was that For example the integral below I would like to know the procedure to rewrite the numerators as (2x-2) + 3 Where does the 3 come from? I would really appreciate Thanks in...
  5. R

    MHB Improper integrals (Comparison Test)

    Use the comparison test to find out whether or not the following improper integral exist(converge)? integral(upper bound:infinity lower bound:2) 1/(1-x^2) dx Here's my solution for 3),but I think something went wrong For all x>=2 0<=-(2-2x)<=-(1-x^2) that means: 0<=-1/(1-x^2)<=-1/(2-2x)...
  6. N

    Problems with limits at infinity within improper integrals

    Homework Statement ##\int_{2}^{\infty} ue^{-u} du## The Attempt at a Solution What I did was find the family of functions described by the indefinite integral ##\int ue^{-u} du## then found the limit as b increases without bound. $$=\lim_{b\rightarrow \infty}...
  7. chisigma

    MHB Integrals with natural logarithm....

    Recently in the 'Challenge Forum' the following integral has been proposed... $$\int_{0}^{\infty} \frac{\ln x}{x^{2}+ a^{2}}\ d x\ (1)$$ Scope of this note is to illustrate a general procedure to engage integrals like (1) in elementary way, i.e. without use comnplex analysis tecniques. The...
  8. Fernando Revilla

    MHB Improper integrals (Comparison Test)

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  9. A

    What lessons should you know before studying integrals?

    so this year I've finshed limits , derivatives (that's it in cacylus)and i'd like to study integrals , ididn't study logarithms yet so idk if that's necessary, thanks
  10. D

    Graphing and Limits for Improper Integrals

    Do you need to know how to graph in order to establish which limit of an improper integrals is going to infinity?? for example: ∫tan(3x)dx from 0 to ∏/6 The integral diverges,, but how do you figure which constant you should use In this problem they put it as the limit a b aproaches...
  11. B

    How Can Abelian Integrals Be Simplified for Beginners?

    Hi guys! Looking at the wiki page for abelian integrals I get no intuition on these scary monsters, & since I'm still not 100% ready as regards all the material in the chapters preceeding sections on abelian integrals in the reference books mentioned on that page I'd think I'd have problems...
  12. B

    Measure theory question on integrals.

    Hi, I was wondering whether if ∫f×g dμ=∫h×g dμ for all integrable functions g implies that f = h? Thanks
  13. A

    MHB Compute Integrals: Integrate (z^3-6z^2+4)dz from -1+i to 1

    Integrate (z^3-6z^2+4)dz where the function is any curve joining -1+i to 1. Z is complex number
  14. W

    Path integrals important for undergrads to know?

    I'm going through a whole undergrad quantum book (Townsend) by myself. It has a chapter on path integral QM. He said in the intro that it can be skipped, but I was wondering if knowledge of this subject is immediately helpful when starting graduate level quantum. I start grad school in the...
  15. S

    MHB Can You Solve These Two Difficult Integrals?

    Prove that \[\int_0^1 \frac{\log(1+x)\log(x)}{1-x}dx=\zeta(3)-\frac{\pi^2}{4}\log(2)\] \[\int_0^1 \frac{\log(1+x^2)}{1+x}dx=\frac{3}{4}\log^2(2) -\frac{\pi^2}{48}\]
  16. G

    Volumes with triple integrals, aka I suck at geometry

    Homework Statement Calculate the volume of the body that is bounded by the planes: x+y-z = 0 y-z = 0 y+z = 0 x+y+z = 2 Homework Equations The Attempt at a Solution I made a variable substitution u = y+z v = y-z w = x which gave me the new boundaries u+w = 2...
  17. J

    Solving Indefinite Integrals: ∫1/(t*ln(t)) & ∫1/(√(t)*[1-2*√(t)])

    Homework Statement ∫1/(t*ln(t)) dt ∫1/(√(t)*[1-2*√(t)]) dt Homework Equations The Attempt at a Solution I used u-substitution for both. For the first equation, my u= ln t, and my final answer was ln|u| + C, or ln(ln(|t|) + C. For the second equation, my u= 1-2*√(t) and...
  18. Lebombo

    Improper Integrals and Series (convergence and divergence)

    Is it safe to say when an integral has an infinite boundary \int_n^∞ a_{n} and the limit yields a finite number, then the integral is said to converge. And when a series has an upper limit of infinity \sum_n^{∞}a_{n} and the limit yields a finite number, then the series is said to diverge.
  19. M

    Finding volume using integrals.

    Huhh First of all I'm sorry if this is the wrong question ( didn't know if this was considered pre cal. I have a gut feeling it is :P) Homework Statement Let R be the region bounded by y=tan x, y=0 and x = Pi/4 Find the volume of the solid whose base is region R and whose cross-section...
  20. M

    Exploring the Relationship Between Line Integrals and Mass in Physics

    Wasn't sure which section to put this q in. Just reading now that f(x,y) can represent the density of a semicircular wire and so if you take a line integral of some curve C and f(x,y) you can find the mass of the wire... makes sense. What I don't get is that if I then move the wire around the...
  21. M

    What is the interpretation of a line integral with a 2D function?

    When I think line integral - I understand when I'm taking a line integral for a function f(x,y) which is in 3D space above a curve that the integral is this curtain type space, just like if you had a 2D function and you find the area under the curve, except now it's turned on its side and it's...
  22. Lebombo

    LCT Limit Comparison Test for Improper Integrals

    Learning about the Limit Comparison Test for Improper Integrals. I haven't gotten to any applications or actual problems yet. Just learning the theory so far, and have a question on the very beginning of it.Homework Statement f(x) ~ g(x) as x→a, then \frac{f(x)}{g(x)} = 1 (that is, f(x)...
  23. P

    Path Integrals- Multivariable Calculus

    Path Integrals-- Multivariable Calculus Hi all-- really stuck here, help would be greatly appreciated. :) 1. Evaluate ∫Fds (over c), where F(x, y, z) = (y, 2x, y) and the path c is de fined by the equation c(t) = (t, t^2, t^3); on [0, 1]: 2. Homework Equations L = sqrt(f'(t)^2 +...
  24. M

    Evaluating Integrals: Additive Interval Property

    Homework Statement Given 7 f(x) dx= 8 0 7 f(x) dx = −3 1 evaluate the following. 1 f(x) dx 0 Homework Equations n/a The Attempt at a Solution I'm a little confused on how to approach this problem. Do i use the additive interval property of integrals?
  25. alyafey22

    MHB Discussions on the convergence of integrals and series

    This thread will be dedicated to discuss the convergence of various definite integrals and infinite series , if you have any question to post , please don't hesitate , I hope someone make the thread sticky. 1- \int^{\infty}_0 \left(\frac{e^{-x}}{x} \,-\,\frac{1}{x(x+1)^2}\right)\,dx\,=1-\gamma...
  26. alyafey22

    MHB Can improper integrals converge without absolute convergence?

    I know we have the following \big | \int^{b}_{a} f(t) \, dt \big | \leq \int^{b}_{a} |f(t)|\, dt 1- How to prove the inequality ,what are the conditions ? 2- Does it work for improper integrals ?
  27. E

    Double integrals, trying to be sure I am not doing something wrong

    1. evaluate the following: \int^{1}_{0}\int^{1}_{0}xyex+y dydxThe Attempt at a Solution OK, so this should be pretty simple. But for some reason I am having trouble integrating the yex+y bit with respect to y. If I do it by parts I end up with iterations like this: \int^{1}_{0}xyex+y -...
  28. W

    Path Integrals Harmonic Oscillator

    Hi, I am reading through the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs and am having a bit of trouble with problem 3-12. The question is (all Planck constants are the reduced Planck constant and all integrals are from -infinity to infinity): The wavefunction for a...
  29. P

    Unlock the Most Challenging Integrals with Jacob

    Hello all, I started a thread exactly like this a few months ago. I really enjoy doing integrals, so if you could post some of the most challenging ones you know of, I would greatly appreciate it :smile: Jacob
  30. D

    Solving Improper Integral: \int_0^{\infty}\frac{1}{x(1+x^2)}

    How can I get the improper integral ## \int_0^{\infty}\frac{1}{x(1+x^2)}\,dx ## First thing I tried was separating the integral like this ## \int_0^{1}\frac{1}{x(1+x^2)}\,dx + \int_1^{\infty}\frac{1}{x(1+x^2)}\,dx## And then I tried with partial fractions but it didn't work
  31. N

    Calculating Triple Integrals in Mathematica

    Homework Statement Evaluate ∫∫∫\sqrt{x^{2} + y^{2}} dA where R is the region bounded by the paraboloid y=x^2+z^2 and the plane y=4 Homework Equations I believe this is a problem where cylindrical coordinates would be useful 0 ≤ z ≤ \sqrt{4-x^2} 0 ≤ r ≤ 2 ( I think this is wrong). 0 ≤ θ ≤...
  32. A

    Improper Integrals: Divergence at x=0?

    Homework Statement Check if the following integrals diverge: Int(1/(x^3-2x^2+x), from 0 to 1) and Int(1/(x^3+x^2-2x) Homework Equations Ratio-test(not sure if that's the name)The Attempt at a Solution I have solved the problem and found that both integrals diverge at x=0. I just want to check...
  33. I

    Double Integrals - Volume of a Cylinder

    Homework Statement A cylinder has a diameter of 2 inches. One end is cut perpendicular to the side of the cylinder and the other side is cut at an angle of 40 degrees to the side. The length at the longest point is 10 inches. Find the volume of the sample. I believe this is what it would...
  34. W

    Line Integral Homework: Integrate (xe^y)ds

    Homework Statement Integrate some area C of (xe^y)ds where C is the arc of the curve x=e^y Homework Equations What is the indeffinite integral and why is it that? Answer is (1/3)e^3y + C The Attempt at a Solution Integral of (xe^y)((e^y)^2 + 1)^(1/2) = Integral of (e^2y)(e^2y...
  35. N

    Use change in variables and iterated integrals theorm to deduce Pappus

    1. Homework Statement [/b] this problem is on page 267 of Advanced calculus of several variables by Edwards, I just can't seem to get a handle on it: Let aA be a contented set in the right half of the xz plane ,x>0. Define $$\hat{x}$$, the x-coordinates of the centroid of A, by...
  36. P

    MHB Interesting ways to evaluate integrals

    Hi everyone. Just for fun I thought we could post some of the more interesting ways we know of to evaluate integrals :) For starters, to evaluate \displaystyle \begin{align*} \int{\arctan{(x)}\,dx} \end{align*}, first we consider the integral \displaystyle \begin{align*} \int{\frac{2x}{1 +...
  37. B

    Integrals with Common Fractions and Area Under Graph

    I'm working on simple game and am working on a leveling system, using a function to get experience needed. I am using area under a function above y=0. The first problem, I can't figure out a simple number. f(x) = x2/5 dx Then, looking for area, I'm unsure about a really simple thing. Getting...
  38. Barioth

    MHB How to Solve These Challenging Integrals with Square Roots?

    Hi everyone I have these 2 integrate that I can't solve, I have tried them with mathematica and wolfram, but they can't find an answer, maybe someone have an idea on how I could tackle these 2 bad boy! The first one is \int{ \sqrt{ \frac{1+( \frac{1}{10}+ \frac{s}{25})^2}{ \frac {s}{10}+...
  39. Petrus

    MHB Double Integrals 2: Solving X Limits

    Hello MHB, I would like to have tips how to solve the x limits for this problem , there Regards,
  40. Petrus

    MHB MHBSolving a Double Integral Problem with Polar Form

    Hello MHB, I got as homework to solve this problem and get recommend to solve it with polar but I have not really work with polar but we have had lecture about it and I have done some research. This is the problem and what I understand \int\int_Dx^3y^2\ln(x^2+y^2), 4\leq x^2+y^2\leq 25 and...
  41. M

    Finding Volume by use of Triple Integrals

    Homework Statement Find the Volume of the solid eclose by y=x^{2}+z^{2} and y=8-x^{2}-z^{2} The Attempt at a Solution Well know they're both elliptic paraboloids except one is flipped on the xz-plane and moved up 8 units. Knowing this, i equated the two equations and got...
  42. Petrus

    MHB Double integrals over general regions.

    Hello MHB, Exemple 3: "Evaluate \int\int_D xy dA, where D is the region bounded by the line y=x-1 and parabola y^2=2x+6" They say I region is more complicated (the x) so we choose y. so if we equal them we get x_1=-1 and x_2=5 then as they said it's more complicated if we work with x limit so...
  43. Petrus

    MHB The double integral of f over rectangle R and midpoint rule for double integrals

    Hello MHB, I wanted to 'challange' myself with solve a problem with midpoint and rule and the double integral f over the rectangle R. This is a problem from midpoint. "Use the Midpoint Rule m=n=2 to estimate the value of the integrab \int\int_r(x-3y^2)dA, where R= {(x,y)| 0\leq x \leq 2, 1 \leq...
  44. S

    Solve Reimann Sums and Integrals

    We are already introduced to finding the value of definite integral by the anti-derivative approach \int_{a}^{b}f(x) dx = F(b) - F(a) In this approach we find the anti-derivative F(x) of f(x) and then subtract F(a) from F(b) to get the value of the definite integral Reimann Sums...
  45. B

    Relationship between Derivatives and Integrals

    Hi, I've recently taken a Calculus 1 (Differential Calculus) course and I've been looking ahead to see what sort of material is covered in the Calculus 2 (Integral Calculus) course. I am wondering about the relationship between derivatives and integrals. From what I understand, an integral...
  46. S

    Calculating integrals involving floor function

    Is it possible to claculate ∫[cot(x)]dx from x=0 to x=\piwhere [.] represents floor functon or the greatest integer function?? it seems impossible to me but can we use the properties of definite integrals to somehow evaluate the area?
  47. N

    Improper integrals and canceling of areas

    When evaluating an improper integral and i get infinity minus infitiy when taking the limit. in what case do the areas cancel?
  48. 6

    Help with Integrals (one of which involves erfc).

    I'm using Green's Functions for heat conduction problems, and I'm trying to solve the following integral: Homework Statement http://img28.imageshack.us/img28/4923/026307b169b04faa8364086.png Where: http://img820.imageshack.us/img820/3742/6332938c445f4b9e8da8ba5.png Homework...
  49. P

    Solving Integrals with Contour Integrals and Cauchy PV

    In trying to solve \int^{\infty}_{-\infty} x + \frac{1}{x} dx could it be split up and solved using the Cauchy Principle Value theorem and a contour integral along a semi-circle. Thus; PV\int^{\infty}_{-\infty}x dx =0 +\int \frac{1}{x} dx = \int^{\pi}_{0} i d\theta Is this valid reasoning?
  50. polygamma

    MHB Integrals: Show $\alpha$ Not Int Multiple of $\pi$ and $s, \lambda >0$

    1) Show that for $\alpha$ not an integer multiple of $\pi$, $\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)} $.2) Show that for $s,\lambda >0$ and $0 \le...
Back
Top