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. S

    Difference Between Surface Integrals and Surface area using double integrals .

    Hi all, Thanks for response :) I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...
  2. C

    Imaginary components of real integrals

    Why does the incomplete gamma function have an imaginary component, when the exponential integral does not? \Gamma(0,z,\infty)\equiv\int^\infty_z \frac{e^{-t}}t dt Ei(z)\equiv-\int^\infty_{-z} \frac{e^{-t}}t dt Looking at how these integrals are usually defined I would have expected them to...
  3. F

    Simple exercise about integrals

    Homework Statement Let ## f: [0, a]## ---> ## \mathbb{R}## be positive and increasing. Prove that the function G, such that: ##G(x):= \frac{1}{x} \int_0^x f(t)dt## ## x\in (0,a)## is increasing. The Attempt at a Solution I know that if the first derivative of a function is positive, that...
  4. R

    Comparison test on second species integrals

    Homework Statement Determine if the following integrals are convergent or divergent. Explain why. \int^{1}_{0} \frac{1}{1-x^{4}} dx The Attempt at a Solution I've tried using Comparison Test, using f(x) = \frac{1}{1-x^{4}} and\; g(x) = \frac{1}{1-x}, 0 \leq f(x)\leq g(x) in ] 0,1 [ and I...
  5. T

    On Taylor Series Expansion and Complex Integrals

    I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my...
  6. B

    Source for calculating Incomplete Elliptical Integrals

    Folks, Is there a online source I can use for evaluating Incomplete elliptical integrals ##F(\phi,k)## and ##E(\phi,k)## I do not have Mathlab or Mathematica and Wolfram alpha requires payed registration for extended computation time. Any information will be appreciated. Regards
  7. P

    Expressing sol. of Poisson eqn. in terms of vol. and sur. integrals.

    Hi, Referring to Jackson's Electrodynamics 3ed, page 197, line 5. He assumes that the magnetization can be divided into volume part and surface part, thus generating eqn 5.100. This is fine. In a straightforward way, I wanted to do the same but for electrostatics, eqn 4.32:∅= (1/4πε) ∫dv...
  8. B

    Surface and Volume Integrals - Limits of Integration

    So I am trying to understand how and why the limits of surface and volume integrals come about. I think I came up with a easy to understand argument but not a mathematically sound one. Frankly its a little dodgy. Can anyone provide feedback on this argument or provide a better and possibly more...
  9. M

    Evalute the following integrals

    Hi Evalute the following integers
  10. C

    Can Gaussian integrals be done with half integrals?

    Is it possible to do Gaussian integrals with half integrals. we would define then nth derivative of e^{-x^2} and then somehow use that. And this integral is over all space. any input will be much appreciated.
  11. J

    Given 2 Integrals, How to solve other Integrals?

    given ∫(2-5) f(x)dx=5 and ∫(4-5) f(x)dx=∏ , find a) ∫(5-5) f(x)dx = b) ∫(5-4) f(x)dx = c) ∫(2-4 f(x)dx = Im going over old tests of mine to get ready for my final, and I can't find anywhere in my notes how I solved this, I originally got (a. 0 b. ∏ c. 5-∏). Can...
  12. Kushwoho44

    Order of Indefinite Double Integrals

    Hi, Rather simple question here, just want to confirm: When we are dealing with indefinite double integrals, it's true to say ∫∫ f(x,y) dx dy = ∫∫ f(x,y) dy dx i.e, order of integration doesn't matter right?
  13. R

    Integrals and exponential growth problem

    Homework Statement Before the AP exam Cal Q Luss has 3 hours to cram: during this time, he wants to memorize a set of 60 derivative/integral formulas. According to psychologists, the rate at which a person can memorize a set of facts is proportional to the number of facts remaining to be...
  14. Vorde

    Work-Energy Theorem with Line Integrals

    Homework Statement The problem is to prove the work-energy theorem: Work is change in kinetic energy.Homework Equations Line integral stuff, basic physics stuff. The Attempt at a Solution I'm given the normal definitions for acceleration, velocity and I'm given Newton's second law. I'm...
  15. A

    A question about Upper Darboux Integrals

    In this link: http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw10-solns.pdf For qustion 32.6, I'm not sure if I'm understanding how there can be a "sequence" of upper and lower darboux integrals. So (for example), what is the difference between U_{10} and U_{11}? Does...
  16. C

    Vector Calculus Question about Surface Integrals

    Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3 BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same...
  17. M

    Conceptual question on greens theorem/line integrals

    Homework Statement Hey guys, I just wanted to know, if it would be an incorrect assumption to say that greens theorem is directly correlated to a line integral. The reason I am assuming that is because the formula for a line integral in my calc text is...
  18. A

    What is the geometrical significance of definite integrals of vector functions?

    What is the geometrical significance of the definite integral of a vector function if any? e.g. if you integrate a vector function that gives the velocity of some particle between t1 and t2, the vector we get indicates the distance traveled in the i, j and k directions right? does the...
  19. D

    Integrals look easy but I'm still confused

    Integrals...look easy but I'm still confused :( Homework Statement evaluate the integral ∫(36/(2x+1)^3)dx Homework Equations dx^n/dx = nx^(n-1) The Attempt at a Solution ∫(36/(2x+1)^3)dx = 6ln[(2x+1)^3]/((2x + 1)^2) ( I know this is wrong, but why??) ∫(36/(2x+1)^3)dx =...
  20. A

    What Are Riemann-Stieltjes Integrals and How Can We Visualize Them?

    I'm having trouble visualizing the riemann-stieltjies integral... Our textbook states: We assume throughout this section that F is an increasing function on a closed interval [a,b]. To avoid trivialities we assume F(a)<F(b). All left-hand and right-hand limits exist...We use the notation...
  21. A

    Change of Variable in multiple Integrals

    Homework Statement Let D be the triangular region in the xy-plane with the vertices (1, 2), (3, 6), and (7, 4). Consider the transformation T : x = 3u − 2v, y = u + v. (a) Find the vertices of the triangle in the uv-plane whose image under the transformation T is the triangle D. (b)...
  22. G

    Integrals going to 0 implies functions go to 0?

    Homework Statement This question is not the assignment problem but I think that if the result I mention here is true, then my assignment problem will be solved. Let (X,\Sigma,\mu) be a measure space. Suppose that (h_n)_{n=1}^\infty is a sequence of non-negative-real-valued integrable...
  23. D

    General solution of double integrals

    I have a differential equation of the form and I want to solve it using calculus, as opposed to using a differential equation method. \frac{d^2v}{dt} = \alpha where v is a function of t i.e., v(t) and \alpha is some constant. How do I solve for v(t) if the time ranges from t_0 to t...
  24. M

    Integrals over a transformed region

    Homework Statement Consider the change of variables x = x(u, v) = uv and y = y(u, v) =u^3+v^3 Compute the area of the part of the x-y plane that is the transform of the unit square in the 2nd quadrant of the u-v plane, which has one corner at the origin. (Since the transformation is 1:1...
  25. alane1994

    MHB How Do You Calculate the Length of a Curve Using Integrals?

    I had a question on a quiz that I missed... I am unsure how they got this answer. If someone could explain it would be great! Write the integral that gives the length of the curve. y=f(x)=\int_{0}^{4.5x} \sin{t} dt It was multiple-choice(multiple-guess;)). \text{Choice A }...
  26. O

    Algorithmic approach to double/triple integrals

    I feel overwhelmed with something that should be capable of being explained very simply I think. Let's say you're getting thrown random questions involving surfaces/shapes creating boundries in ℝ3. Whats your step by step process in finding whether you want to do a double integral versus...
  27. U

    Integrate curve f ds Line Integrals

    Homework Statement Compute ∫f ds for f(x,y)= √(1+9xy), y=x^3 for 0≤x≤1 Homework Equations ∫f ds= ∫f(c(t))||c'(t)|| ||c'(t)|| is the magnitude of ∇c'(t) The Attempt at a Solution So, with this equation y=x^3 ... I got the that c(t)= <t,t^3> c'(t)=<1,3t^2> I know that from the equation...
  28. N

    Find the exact volume of a bounded surface. Multiple Integrals.

    1. Ok, so the question is.. Find the exact volume of the solid bounded above by the surface z=e^{-x^2-y^2}, below by the xy-plane, and on the side by x^2+y^2=1. 2. Alright. So, I know that I can use a double integral to find the volume, and switching to polar coordinates would be simpler...
  29. M

    Double integrals using polar co-ordinates

    Homework Statement Step 1) I put the following into polar coordinates √(16-x2-y2)=√16-r2 Where r≤4 step 2 I solved for y in the original problem which is in the link y≤√(4-x2) step 3. I graphed the above function step 4. I put the above function in polar coordinates...
  30. L

    Interchange of limits and integrals

    Hi, I was wondering about one particular example of this interchange. In Mallat's book, at the proof of Poisson Formula it's visible that the equation at the beginning of the 42nd page features the limit outside of the integral. It is my understanding that this limit had to be in...
  31. F

    Solving Trigonometric integrals using cauchy residue theorem

    Homework Statement evaluate the given trigonometric integral ∫1/(cos(θ)+2sin(θ)+3) dθ where the lower limit is 0 and the upper limit is 2π Homework Equations z = e^(iθ) cosθ = (z+(z)^-1)/2 sinθ = (z-(z)^-1)/2i dθ = dz/iz The Attempt at a Solution after I substitute and...
  32. M

    Double integrals using polar co-ordinates

    Homework Statement ∫∫e-(x^2+y^2) dA R Where R is the region enclosed by the circle x2+y2=1 First thing I did was graph the region where the function was enclosed. I saw that they didnt give a limitation to where the circle lied. So I automatically knew that d(theta) would lie on the...
  33. R

    Integrals of Exponential(Polynomial(x)) dx Form

    I'm curious about the general solution to \int_{-\infty}^{+\infty} \exp[P(x)] dx Where P(x) is a polynomial in x with real coefficients and whose leading (highest) order is even and its leading order coefficient is negative. Intuitively these integrals ought to converge, but I'm having...
  34. E

    Definite integrals: solving with residue theory and contour integration

    Homework Statement I need to solve this integral for a>0: \int _0^{\infty }\frac{\text{Sin}[x]}{x}\frac{1}{x^2+a^2}dx The Attempt at a Solution Using wolfram mathematica, I get that this integral is: \frac{\pi -e^{-x} \pi }{2x^2}=\frac{\pi (1-\text{Cosh}[a]+\text{Sinh}[a])}{2...
  35. R

    Interpreting path integral averages as measure integrals

    Hi all, Sorry if this is in the wrong place. I'm trying to understand probability theory a bit more rigorously and so am coming up against things like lebesgue integration and measure theory etc and have a couple of points I haven't quite got my head around. So starting from the basics...
  36. S

    How Does a Vector Field Affect a Projectile's Path?

    So I was wondering if I defined a vector field F, and a Trajectory of a particle x=t y=.5at^2+vit+si and I can find the work done by the field on a particle moving on a path with a line integral ∫F.dr, so what would this equate to for a projectile does it apply to this?, could you give me a real...
  37. C

    How Do You Calculate Error Bounds for Maclaurin Polynomial Approximations?

    Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam... Let's say: I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5 The 5th degree polynomial is 1 + x^2 + (1/2)(x^4) So my queries are: How would I go about...
  38. S

    Laurent Series-Finding Contour Integrals

    Homework Statement Evaluate ∫f(z)dz around the unit circle where f(z) is given by the following: a) \frac{e^{z}}{z^{3}} b) \frac{1}{z^{2}sinz} c) tanh(z) d) \frac{1}{cos2z} e) e^{\frac{1}{z}} Homework Equations This is the chapter on Laurent Series, so I'm pretty sure...
  39. M

    Evaluate integrals using modified Bessel function of the second kind

    Hi guys, I encountered it many times while reading some paper and textbook, most of them just quote the final result or some results from elsewhere to calculate the one in that context. So I'm not having a general idea how to do this, especially this one \int_k^\inf...
  40. R

    Finding Whether Improper Integrals Converge

    Hi all, I'm having trouble with finding an improper integral. The problem is ∫10(xln(x))dx My book says the answer is -1/4, but I do not understand how this is the case. lim(xlnx) as x approaches 1- = 0 lim(xlnx) as x approaches 0+ = ∞ So how does this value converge at -1/4? Thanks in...
  41. G

    Using complex numbers for evaluating integrals

    How can I use complex numbers to evaluate an integral? For instance I'm reading a book on complex numbers and it says that to evaluate the integral from 0 to pi { e^2x cos 4x dx }, I must take the real part of the integral from 0 to pi { e^((2 + 4i)x) dx}. It totally skips how you do that. I...
  42. J

    MHB What is the method for integrating 1/(1+x^n) using roots of polynomials?

    $(1)\displaystyle \;\; \int\frac{1}{1+x^6}dx$ $(2)\displaystyle \;\; \int\frac{1}{1+x^8}dx$
  43. C

    Double integrals over finite region

    Homework Statement Evaluate \int_{R} \int \frac{xy^2}{(4x^2 + y^2)^2} dA where R is the finite region enclosed by y = x^2\,\,\text{and}\,\, y = 2x The Attempt at a Solution I think the easiest way to integrate is to first do it wrt x and then wrt y, i.e \int_{0}^{4}...
  44. R

    Numerical Integration of Double integrals

    I have a double integral: ∫∫sin^2(∏x/A)*sin^2(∏y/B)dxdy A=length along x B=length along y ranges: 0 to A(for x) & 0 to B (for y) Analytical result is: A*B/4 (unit^2) Now, I want to evaulate it numerically using trapezoidal rule. Infact, I have done it but not sure whether it is a right...
  45. L

    Phase Space, Velocity, Integrals

    Hi, this is my first post. I did a search and in this sub-forum I found the most related threads for what I'm looking for. I need some guidance or where or how to learn all this mathematics for velocity-phase space integrals that appear in Maxwellian distributions. I'm an Engineer in...
  46. majormaaz

    Acceleration of a brick, involving integrals

    Homework Statement A 15 kg brick moves along an x axis. Its acceleration as a function of its position is shown in Fig. 7-32. What is the net work performed on the brick by the force causing the acceleration as the brick moves from x = 0 to x = 8.0 m? Homework Equations W = FΔx = max...
  47. D

    Finding the amount of work done (line integrals)

    Homework Statement Find the amount of work (ω) done by moving a point from (2;0) to (1;3) along the curve y=4-(x^2), in the effect of force F=(x-y;x). Homework Equations The Attempt at a Solution ω = ∫((x-y)dx + xdy) ω = ∫(x-4+x^2)dx + ∫√(4-y) dy In the end, I get this...
  48. Kawakaze

    Help with revision - area integrals

    I came across this question on a past paper and would appreciate some help. It is too hard for me at the minute. The problem - The volume of a body whose surface is formed on the underside by the paraboloid z = x2 + y2 and bounded on top by the cone z = 3-2(x2 +y2) (a) Explain which...
  49. D

    Fundamental theorem for line integrals

    Hi, I have a question. In my calculus book, I always see the fundamental theorem for line integrals used for line integrals of vector fields, where f=M(x,y)i + N(x,y)j is a vector field.The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is...
  50. phosgene

    Derivatives of integrals and inverse functions

    Homework Statement Find the derivative of: 1. f(x)=arccos(5x^3) 2. f(x)=∫cos(5x)sin(5t)dt when the integral is from 0 to x Homework Equations Chain rule, dy/dx=dy/du*du/dx The Attempt at a Solution For the first one, I can just take 5x^3 as u and then apply the chain rule...
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