What is Integral: 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. E

    Why is the integral of a(x) different from the integral of a(t)?

    Hello! First time poster, please treat me well! :wink: I've already solved the problem below on my second attempt with the help of kinetic energy but I want to know why my first attempt gives a wrong answer. 1. Homework Statement A force in the +x-direction with magnitude F(x) = 18.0 N -...
  2. Math Amateur

    MHB Understanding Irreducible Elements in Integral Domains - Peter's Questions

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  3. Math Amateur

    I Definition of an irreducible element in an integral domain

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  4. F

    A On the equivalence of operator vs path integral in QFT

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  5. P

    A Closed form for series over Exponential Integral

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  6. D

    A How to Solve the Fourier Integral in Eq. (27) Involving Position Vectors?

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  7. X

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  8. Oedipus

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  9. J

    I Multi-dimensional Integral by Change of Variables

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  10. Mr Davis 97

    I Integrating a Complex Integral using Substitution and Simplification

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  11. Math Henry

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  12. Isaac0427

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  13. P

    I Solve Challenging Integral with Proven Techniques | x>1 Integer Solution

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  14. J

    How can I find this surface integral in cylindrical coordina

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  15. D

    Calculus Calculus of variation textbook 'not under a single integral'

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  16. TheSodesa

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  17. K

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  18. ShayanJ

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  19. N

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  20. binbagsss

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  21. grandpa2390

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  22. O

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  23. T

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  24. R

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  25. R

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  26. kolleamm

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  27. binbagsss

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  28. substitute materials

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  29. I

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  30. Anne Leite

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  31. A

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  32. Alettix

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  33. Rectifier

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  34. Rectifier

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  35. Rectifier

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  36. Rectifier

    Integral Inequality: Prove x-1 > Int(sin(t)/t) for x>1

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  37. A

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  38. Rectifier

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  39. Rectifier

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  40. P

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    Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.
  41. mamadou

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  42. F

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  43. karush

    MHB Integral of a rational function

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  44. Rectifier

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  45. J

    MHB Integral of Fresnel functions

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  46. bwest121

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  47. ShayanJ

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  48. BiGyElLoWhAt

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  49. BiGyElLoWhAt

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  50. Rectifier

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