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. Pull and Twist

    MHB Volume w/ Double Integrals: What Am I Doing Wrong?

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  2. D

    Special Integrals Hermite(2n+1,x)*Cos (bx) and e^(-x^2/2)

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  3. Byeonggon Lee

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    I only memorized these trigonometric differential identities : `sin(x) = cos(x) `cos(x) = -sin(x) because I can convert tan(x) to sin(x) / cos(x) and sec(x) to 1 / cos(x) .. etcAnd there is no need to memorize some integral identities such as : ∫ sin(x) dx = -cos(x) + C ∫...
  4. Breo

    Why to change from momentum space integrals to spherical coordinate ones?

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  5. S

    Are These Triple Integrals Set Up Correctly?

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  6. YogiBear

    Finding a parametric form and calculating line integrals.

    Homework Statement Let C be the straight line from the point r =^i to the point r = 2j - k Find a parametric form for C. And calculate the line integrals ∫cV*dr and ∫c*v x dr where v = xi-yk. and is a vector field Homework EquationsThe Attempt at a Solution For parametric form (1-t)i + (2*t)j...
  7. O

    Force in Varying Places in a Swimming Pool

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  8. U

    What Do Different Types of Integrals Represent in Calculus?

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  9. M

    Integrals of the function f(z) = e^(1/z) (complex analysis)

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  10. kostoglotov

    Double Integrals: Where am I making a mistake?

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  11. F

    Splitting Fractions (Integrals)

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  12. P

    Symmetry in Integrals: Peskin's Equation 6.43 & 6.44

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  13. F

    Find Work Done Using Two Different Integrals

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  14. RJLiberator

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  15. B

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  16. U

    What Does the Double Integral ∫ ∫ f(x,y) dx dy Represent?

    I am just starting to learn double and triple integrals. Say: I = ∫ ∫ dx dy This should give the area within two curves right? What will the following integral give? Will it give volume? I am finding questions where it gives (for instance) mass in a given shape I = ∫ ∫ f(x,y) dx dy
  17. B

    Divergence and Volume Integrals

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  18. Calpalned

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  19. Calpalned

    Can Single, Double, and Triple Integrals Vary in Variable Count?

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  20. O

    Difficult Question in Calculus — limits and integrals

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  21. M

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  22. geezer73

    Double Integrals in Polar Coordinates

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  23. L

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  24. F

    Feynman Path integrals in space with holes?

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  25. andyrk

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  26. titasB

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  27. M

    How to solve for first integrals of motion

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  28. L

    How do I normalize a wavefunction in three dimensions?

    Homework Statement 2. Homework Equations [/B] Uploaded as a picture as it's pretty hard to type out The Attempt at a Solution So to normalise a wavefunction it has to equal 1 when squared. A is the normalisation factor so we have: A.x2e-x/2a0.x2e-x/2a0 = 1 ∫ψ*ψdx = A2∫x4e-axdx = 1 Then I've...
  29. ShayanJ

    How to Transform Integrals from Cylindrical to Spherical Coordinates?

    Consider an integral of the type ## \int_0^{a} \int_0^{\pi} g(\rho,\varphi,\theta) \rho d\varphi d\rho ##. As you can see, the integral is w.r.t. cylindrical coordinates on a plane but the integrand is also a function of ##\theta## which is a spherical coordinate. So for evaluating it, there are...
  30. A

    Surface Integrals: Understanding & Examples

    Homework Statement Its more of a general issue of understanding than a specific problem I have to evaluate a few surface integrals and I am not sure about the geometric significance of what I am evaluating or even of what to evaluate. Examples. If n is the unit normal to the surface S...
  31. J

    Improper integrals: singularity on REAL axis (complex variab

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  32. M

    Unitarity and locality on patgh integrals

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  33. E

    Surface area - Double integrals

    Hi! Here is my task: Calculate surface area of sphere $$x^{2}+y^{2}+z^{2}=16$$ between $$z=2$$ and $$z=-2\sqrt{3}$$. Here are 3D graphs of our surfaces: Surface area of interest is P3. It would be P-(P1+P2), where P is surface area of whole sphere. Is it correct? Here is how I calculated...
  34. K

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    Homework Statement Here are the problems http://imgur.com/a/kbtPS The problems I need help with are 1 and 4(a) Homework Equations The second fundamental theorem of calculus The Attempt at a Solution For problem 1, I calculated the areas under the curve (using remmien summs) and tried to find...
  35. K

    Solve for y(x) using the Fundamental Theorem of Calculus

    Homework Statement Solve the integral equation for y(x): y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt (integral from 0 to x) See attached image for the equation in a nicer format. Homework Equations Fundamental Theorem of Calculus The Attempt at a Solution dy/dx = y(x)^2 / (1 + x^2) ∫ dy/y^2 = ∫...
  36. B

    Surface Integrals and Gauss's Law

    When I learned Integrals in Calc III, the formula looked like this ∫∫ F(r(s,t))⋅(rs x rt)*dA but in physics for Gauss's law it is ∫∫E⋅nhat dA How are these the same basic formula? I know that nhat is a unit vector, so it is n/|n|, but in the actual equation, it is a dot between the cross...
  37. Cake

    Integrals with undefined bounds

    Homework Statement Find the area enclosed by the equations: y=1/x and y=1/x^2 and x=2 Homework Equations N/A The Attempt at a Solution So I solved this analytically after looking at a graph of the two functions. Using integrals I got the following: ln(2)-1/2 Which is the correct answer. I...
  38. P

    Leibniz rule for double integrals

    Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...
  39. A

    How to deal with this sum complex analysis?

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  40. T

    A little confused about integrals

    I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function? If an integral is the area under a curve, why do we even have to find the anti derivative...
  41. andyrk

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    I had a couple of questions. 1. Why does the integral ∫exf(t) dt transform to ex∫f(t) dt? Shouldn't ex be a part of the integrand too? 2. Why is the difference dy - dy1 = d(y - y1)?
  42. R

    MHB Three indefinite integrals involving e^x

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  43. andyrk

    Continuity in Integrals and Antiderivatives

    I was a bit confused by the definition of integrals (both definite and indefinite) and anti-derivatives. The definition for indefinite integrals is- The indefinite integral of a function x with respect to f(x) is another function g(x) whose derivative is f(x). i.e. g'(x) = f(x) ⇒ Indefinite...
  44. P

    The Difference Between Two Indefinite Integrals

    I actually came across this question on social media. What is: $$\int sin (x) \, dx - \int sin (x) \, dx$$ And I think the answer depends on how we interpret: $$\int sin (x) \, dx$$ If we think of it as a single antiderivative, the answer would be zero. If we think of it as being...
  45. M

    Integrals and gamma functions manipulation

    Homework Statement I am working through some maths to deepen my understanding of a topic we have learned about. However I am not sure what the author has done and I have copied below the chunk I am stuck on. I would be extremely grateful if someone could just briefly explain what is going on...
  46. A

    Explain this method for integrals (complex analysis)

    I saw this method of calculating: $$I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx$$ http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx Can you take a look at M.N.C.E.'s method? I don't understand a few things. Somehow he makes the relation...
  47. A

    Proving integral on small contour is equal to 0.

    Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ $R$ is the big radius, $\delta$ is the small radius. Actually, let's consider $u$ the small radius. Let $\delta = u$ Ultimately the goal is to let $u \to 0$ We can parametrize, $$z =...
  48. A

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  49. T

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  50. H

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