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

    Question about Indefinite integrals

    Homework Statement I hope this is in the right forum, because this is a question on theory and not related to a specific problem. I was reading onlne about the Fundamental Theorem of Calculus. On one site the author wrote: F(x) = \int_{0}^{x} f(t) dt Later, he wrote: \int_{a}^{b}...
  2. B

    Intuition of improper integrals of type II

    Let's say you have a function that is continuous in (a,b] but discontinuous at x=a and you integrate it from a to b. For example, \int^{1}_{0} \frac{1}{\sqrt{x}}dx I understand that the integral exists, and it can be easily computed by using the limit as x approaches 0 from the positive...
  3. D

    MHB Solving Even Function Integrals: $a_0$ & $a_n$

    $f(-\theta) = |\sin(-\theta)| = |-\sin\theta| = |\sin\theta| = f(\theta)$. Hence, $f$ is even, and we need to only consider. $$ f(\theta) = \frac{a_0}{2} + \sum_{n = 1}^{\infty}a_n\cos n\theta. $$ $$ a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi}|\sin\theta|d\theta = \frac{1}{\pi}\int_{0}^{\pi}\sin\theta...
  4. B

    Questions about definite integrals

    While reading my calc book, I have developed a few questions about the situations in which definite integrals can exist. I've thought about these questions, and I feel that if I am able to answer some of them, I can make some other problems much easier, such as testing for convergence of a...
  5. M

    Error Analysis and Simpson's Rule to approximate integrals

    Hey there, This isn't exactly a homework question, but my question did arise while doing homework and I wouldn't know where else to post it, so here I am. I'm a Calc 2 student and covered approximations of integrals that I normally wouldn't be able to integrate today. I was going through the...
  6. S

    Integrals - the Substitution Rule with sin^n(x)

    Homework Statement Given that n is a positive integer, prove ∫sin^n(x)dx=∫cos^n(x)dx from 0 -> pi/2 Homework Equations Perhaps sin^2(x)+cos^2(x)=1? Not sure. The Attempt at a Solution I honestly don't even know where to start. Should I set u=sin(x) or cos(x)? Doesn't seem to get...
  7. M

    Numerical computation of Oscillatory integrals

    Hi all, I have a family of nasty integrals to do, all of the general form \int ^{\infty} _0 db b J_0 (b q)(e^{i\chi(b)}-1) where chi is a complicated real function, typically a sum of products of special functions. (q is a real number.) The only thing that the different functions have...
  8. C

    Double Integrals - application to centre of mass.

    I know that for a system of particles, the x, y and z coordinates of the centre of mass are given by (\frac{1}{M} \sum_{i = 1}^{n} m_i x_i, \frac{1}{M} \sum_{i = 1}^{n} m_i y_i,\frac{1}{M} \sum_{i = 1}^{n} m_i z_i). For a solid body, we can treat this like a continuous distribution of matter...
  9. I

    Does anyone know a resource for advanced Methods for ODEs, Integrals, etc.

    Is there a resource that is just a walkthrough of various kinds of problems one might get and the ways to solve them? I'm not talking about the basics from the calc and difEQ series (u substitution, partial fraction decomposition, trig substitutions, trig power reduction, integration by parts...
  10. P

    What Is an Integral in Simple Terms?

    Hello, I recently learned about integrals in my high school calculus class. Yes, I know you add 1 to the exponent and divide the coefficient by that number, and the integral of a velocity over time graph gives the total distance traveled, but I have a different question: What is a quick and easy...
  11. B

    Solving Line Integrals with Vector Cross Products

    I posted an actual problem in advanced physics but no answer so i will try to get an math part answer from it. Suppose I have to solve this integral: I=\int {\vec{dl} × \vec A } Where \vec A = -\frac {1}{x} \vec a_{z} So it has only a z component and I have to find the vector cross of the...
  12. B

    Force on triangular current loop: line integrals

    Homework Statement http://pokit.org/get/img/65e8ba92c1d00bf7fc8be2b178757ed8.jpg If a=5b, and I1 and I2 are known, find the force on the triangular loop. Homework Equations The Attempt at a Solution For start, the field from the infinitely long wire is : \vec B=\large -\frac{\mu _{0}...
  13. J

    Exploring Why Treating Variables as Constants Fails in Line Integrals

    Hey guys and gals, this isn't actually an assignment of any sort, so I didn't want to put it in the homework section. This is also my first post, though I have been lurking for quite a while, reading the copious amounts of information available here. :p Anyhow, could somebody please elaborate...
  14. U

    Change of Variables in Tripple Integrals

    Homework Statement In double integrals, the change of variables is fairly easy to understand. With u = constant and v = constant, along line KL v = constant so dv = 0. Therefore the only contributing variable to ∂x and ∂y is ∂v. The Attempt at a Solution However, in tripple...
  15. G

    Contour Integrals: Calculate \oint_{C} (y^2+ix)dz

    Homework Statement calculate the contour integral \oint_{C} (y^2+ix)dz where C consists of the parabolic path z(t)=t^{2}+it for 0≤t≤1 followed by the straight line segment from the point 1+i to the point 0 Homework Equations The Attempt at a Solution so the contour is in 2 parts...
  16. M

    Dirac notation expressions as integrals

    Can anyone point me to how to interpret Dirac notation expressions as wave functions and integrals beyond the basics of     <α| = a*(q)     |β> = b(q)     <α|β> = ∫ a* b dq For example in the abstract Dirac notation the expression     |ɣ> (<α|β>) can be evaluated as     (|ɣ><α|) |β>  ...
  17. A

    A question about Feynman's book of Path. integrals

    Page 35 Feynman Quantum Mechanics and Path Integrals (2-26)ψ=(iε)R I know it's a defination.But how to find some threads.How can we get this?
  18. S

    Are Integration and Differentiation True Opposites in Calculus?

    What I have learned in school is that differentiation and integration are opposites. By integrating a function we find the area under the graph. So, integration gives us the area. Differntiation gives slope of the function. If I am right by saying differentiation and integration are...
  19. D

    Derivative of the product of 2 definite integrals

    Homework Statement Find f'(x) for (integral from 0 to x of cos^5(t)dt)* (integral from x^2 to 1 of e^t^2 dt). No differentiation allowed in the answer Homework Equations The Attempt at a Solution. I used the product rule and integrated then differentiated the first term --> cos^5x*...
  20. S

    Guessing integrals - change of variable

    Homework Statement I'm working through Mahajan's Street-fighting Mathematics for fun, and am a bit puzzled about the following problem: The Attempt at a Solution It's mostly the first part that's tripping me up. I could use the substitution tan\theta=x and solve both integrals...
  21. L

    How to solve these indefinite integrals of composite functions

    Homework Statement Well I have these three different integrals: \int{\frac{1}{\sqrt{4x^2-1}}dx} \int{\frac{1}{4-x^2}dx} \int{\frac{1}{x^2+4x+8}dx}Homework Equations Yeah well not exactly sure how to approach this... Do you use integration by substitution, where you come up with some...
  22. T

    Surface integrals evaluation problem

    Homework Statement Evaluate ∫∫F.nds where F=2yi-zj+x^2k and S is the surface of the parabolic cylinder y^2=8x in the first octant bounded by the line y=4, z=6 Homework Equations We were told that the projection is supposed to be taken in the yz plane but how?? and i have a feeling that...
  23. K

    Why do derivatives and integrals cancel each other?

    Is there any clear explanation as to why exactly derivatives and integrals cancel each other [other than the integral is the anti-derivative]? To my understanding the derivative gives the slope of a curve at given point whereas the the integral finds the area under the curve. How are these...
  24. C

    Straight Question, Spivak, Integrals

    Homework Statement Exercice 14-14, 3^{rd} edition, Spivak: Find a function f such that f^{```}(x) = \frac{1}{\sqrt{1 + sin^{2}x}}. The answer acording to the book is : f(x) = \int\left(\int\left( \int \frac{1}{\sqrt{1+sin^{2}x}} dt \right) dz...
  25. F

    Setting up Triple Integrals over a bounded region

    Homework Statement Set up triple integrals for the integral of f(x,y,z)=6+4y over the region in the first octant that is bounded by the cone z=(x^2+y^2), the cylinder x^2+y^2=1 and the coordinate planes in rectangular, cylindrical, and spherical coordinates. Homework Equations...
  26. D

    Can Definite Integrals Be Divided Algebraically?

    Is there a general algebraic way to write the quotient of two definite integrals as one? I mean, what would be \frac{\int_a^b f(s) ds}{\int_c^d g(t) dt} Is it analogous to the product of integrals creating a double integral? Thanks in advance!
  27. N

    Proper mathematical notation in regarding integrals

    Hi I have a question regarding notation. I have a function f(x, y), which I would like to integrate as \int_{x>0,y<x\frac{1}{\sqrt{\pi}}+1}{f(x, y) dxdy} My question is very simple, and probably very silly: What there a notation which enables me to write the condition for the integral (x>0, y...
  28. C

    How to Calculate Definite Integrals?

    I made a pdf so that the equation would be more clear
  29. D

    Finding the Product of Integrals

    Is there a formula for calculating the product of integrals, something like: \left(\int_a^b f(x) dx\right) \times \left(\int_c^d g(y) dy\right) when there is no closed-form expression for F(x) and G(y). Actually, the functions are almost identical, f(x) = x^p e^{-x} \text{ and }...
  30. S

    MHB Evaluating Integrals Involving Trig Functions

    Evaluate: 1. $\displaystyle \int_0^{\displaystyle 2\pi} \frac{x \sin^{2n}(x)}{\sin^{2n}(x)+\cos^{2n}(x)}dx$, $n>0$ 2. $\displaystyle \int_0^{ \displaystyle \pi \over \displaystyle 2} \frac{x \sin x \cos x}{\sin^{4}(x)+\cos^{4}(x)}dx$
  31. A

    MHB Finding volumes by multiple integrals

    How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me. Find the volumes in R3. 1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane surfaces z=2, x+z=1. 2. Find the volume W that is bounded by the cylindrical...
  32. A

    Multiple integrals for finding volume

    How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me. Find the volumes in R3. 1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane surfaces z=2, x+z=1. 2. Find the volume W that is bounded by the cylindrical...
  33. F

    Upper and lower Riemann integrals of The Riemann Stieltjes Integral

    Homework Statement Homework Equations The Attempt at a Solution This is what I got so far. I haven't seen any similar problems like this, and this is my first attempt here. I wonder if I did it right. Can anyone check this for me? If you need additional explains or...
  34. A

    Triangles and Definite Integrals

    I'm trying to figure out how to integrate a data set, without knowing the function. While doing this, I got to thinking about this: If the definite integral of a function can be represented by the area under that function, bound by the x axis, then shouldn't: \int_{a}^{b}2x\frac{\mathrm{d}...
  35. 6

    How to Solve Equations with Integrals Using FTC and Chain Rule?

    Hi, How would one go about solving equations like ∫^{b}_{a}f(s,t)g(s)ds=g(t),for f(s,t). Could we turn it into a differential equation somehow? Thanks
  36. A

    Equality in the Cauchy-Schwarz inequality for integrals

    Homework Statement Regarding problem 1-6 in Spivak's Calculus on Manifolds: Let f and g be integrable on [a,b]. Prove that |\int_a^b fg| ≤ (\int_a^b f^2)^\frac{1}{2}(\int_a^b g^2)^\frac{1}{2}. Hint: Consider seperately the cases 0=\int_a^b (f-λg)^2 for some λ\inℝ and 0 < \int_a^b (f-λg)^2 for...
  37. I

    Problem with the change of variable theorem of integrals

    Homework Statement So I came across the integral \int^{1}_{0}x\sqrt{1-x^2} and I tried to solve it in two ways using the change of variables theorem for integration, however both ways are supposed to give me the same result, but they differ in the sign and I cannot find what I am doing...
  38. N

    Regions of integration; bounds and triple integrals.

    Homework Statement My first problem is with 2ia) and 2ib), I got the correct answer, although not happy with my understanding of it. http://img826.imageshack.us/img826/1038/443pr.jpg The Attempt at a Solution (2ia and 2ib) The region that's of concern is the upper part between y = x^2 and...
  39. M

    Logarithmic and exponential integrals

    In this text, I will ask a question about the power series expansion of exponential and logarithmic integrals. Now, to avoid confusion, I will first give the definitions of the two: \mathrm{Ei}(x)=\int_{-\infty}^{x}\frac{e^t}{t}dt \mathrm{Li}(x)=\int_{0}^{x}\frac{dt}{\log(t)} where Ei denotes...
  40. T

    Triple Integrals: Spherical Coordinates - Finding the Bounds for ρ

    Homework Statement Find the volume of the solid that lies above the cone z = root(x2 + y2) and below the sphere x2 + y2 + x2 = z. Homework Equations x2 + y2 + x2 = ρ2 The Attempt at a Solution The main issue I have with this question is finding what the boundary of integration is for ρ. I...
  41. A

    Integrals as the area under a curve

    I have always seen the integral as the area under a curve. So for instance, if you integrated over the upper arc of a circle you would get ½\piR2. But then, I learned to do integrals in spherical coordinates and something confuses me. If you do the integral from 0 to 2\pi, you don't get the...
  42. T

    Triple Integrals: Finding Mass of a Bounded Solid

    Homework Statement Find the mass of a solid of constant density that is bounded by the parabolic cylinder x=y2 and the planes x=z, z=0, and x=1. The Attempt at a Solution https://dl.dropbox.com/u/64325990/Photobook/Photo%202012-06-07%202%2033%2024%20PM.jpg I first drew some diagrams to...
  43. T

    Applications of Double Integrals: Centroids and Symmetry

    Homework Statement A lamina occupies the region inside the circle x2+y2=2y but outside the circle x2+y2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin. Here is the solution...
  44. L

    Miscellaneous Definite Integrals

    Homework Statement show that \int^{∞}_{0}\frac{sin^{2}x}{x^{2}}dx= \frac{\pi}{2} Homework Equations consider \oint_{C}\frac{1-e^{i2z}}{z^{2}}dz where C is a semi circle of radius R, about 0,0 with an indent (another semi circle) excluding 0,0. The Attempt at a Solution...
  45. M

    MHB Can You Help Me Solve This Trigonometric Integral?

    Hello again. I am hoping that someone can assist in checking my work regarding a trigonometric integral. The problem and my attempt to solve is as follows. \int\sin^{\frac{-3}{2}}(x)*cos^3(x) dx \int\sin^{\frac{-3}{2}}(x)*cos^2(x)*cos(x) dx Using a Pythagorean identity...
  46. T

    Double Integrals: Will this solution always give the correct answer?

    Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/Photobook/question.PNG Here is the answer: http://dl.dropbox.com/u/64325990/Photobook/solution.PNG So what the answer says is that you can find the volume under the surface minus the volume of the rectangle with height...
  47. ShayanJ

    Differentiation of integrals and integration(?)

    I heard that the formula below can be used to evaluate some kinds of integrals but I can't find what kinds and how to do it.Could someone name those kinds and also the procedure? \frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) b'(x) - f(x,a(x)) a'(x) + \int_{a(x)}^{b(x)}...
  48. C

    Difference between double and surface integrals? Purpose of surface integrals?

    I'm preparing for a vector calculus course in the fall and I've been self studying some topics. I've taken multivariable calculus and I'm familiar with using double integrals, how to solve them and how to use them to find volume. From what I've read so far, I'm familiar with how to SOLVE a...
  49. R

    Measuring volume of spheres using triple integrals

    Homework Statement I'm just interested in knowing where the 4 comes from in front of the integral.
  50. R

    Converting cartesian to polar coordinates in multiple integrals

    Homework Statement Do you see how y gets converted to csc? I don't get that. I would y would be converted to sin in polar coordinates.
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