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.

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

    If you do not answer the above questions, you will not have a correct answer.

    Homework Statement Determine the area of the surface A of that portion of the paraboloid: [x][/2]+[y][/2] -2z = 0 where [x][/2]+[y][/2]≤ 8 and y≥x Homework Equations Area A = ∫∫ dS The Attempt at a Solution Area A = ∫∫ dS = 3∫∫ dS
  2. B

    Calculating Integral Using Residue Theorem & Complex Variables

    Homework Statement I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...
  3. K

    Calculating Line Integral in xy-Plane

    Homework Statement Calculate the line integral ° v ⋅ dr along the curve y = x3 in the xy-plane when -1 ≤ x ≤ 2 and v = xy i + x2 j. Note: Sorry the integral sign doesn't seem to work it just makes a weird dot, looks like a degree sign, ∫.2. The attempt at a solution I have to write something...
  4. dRic2

    I "Imagine" a definite integral with a finite number of discontinuties

    Hi, I'm re-studying integrals and I got stuck with this problem. Actually the math beyond it is very clear but I still can figure it out. Take this function: ##f(x) = \begin{cases} 0, & x \lt 1 \\ 1, & x = 1 \\ 0, & x > 1 \end{cases} ## According to Spivak's Calculus I, a function is...
  5. D

    What is the mistake in calculating the integral of the absolute sine function?

    Homework Statement \int_0^{2018 \pi} \lvert \sin(2018x) \lvert \mbox{d}x Homework EquationsThe Attempt at a Solution So the period is: \frac{2 \pi}{ 2018} Each "hump" of the sine has an area of 2 so if I count the number of humps I am done. In one period of an absolute sine function the...
  6. T

    Potential Difference Problem - setting up the integral

    Homework Statement Homework Equations V=kq/x The Attempt at a Solution I know the correct solution. It's... On my first attempt, rather than use (d+x) in the denominator and integrate from 0 to L, I instead used (x) and integrated from (d) to (L+d). This produces the wrong answer...
  7. Phylosopher

    I Deriving a function from within an integral with a known solution

    Hello,I am not sure if these types of problems are Intermediate or advanced. I am not sure too whether they have a certain name or not. I have a function inside a definite integral. The solution of this definite integral is known. What is the function that satisfy the known solution. In...
  8. J

    Mathematica Strange Integral Results: Is Something Wrong?

    As shown in the image below, I tried to integrate a large integral. However, the result is strange. According to the result, the integral is always zero whatever the values of w, h, L, P, S and k. However, when I try to put some "test values", the result is not zero. test values...
  9. Euge

    MHB POTW: Evaluate Integral for Positive Integer n

    Here is this week's POTW: ----- If $n$ is a positive integer, evaluate $$\int_{0}^\infty \frac{dx}{1 + x^n}$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...
  10. E

    I What Steps Are Needed to Solve This Integral Problem?

    Hi, I am trying to solve this integral: \int_0^{\infty}\frac{\ln(1+\alpha\,x)}{(1+x)^2}\,dx Using integration by parts this can be written as: -\frac{1}{1+x}\ln(1+\alpha\,x)\Big|_0^{\infty}\Big. + \alpha\int_0^{\infty}\frac{1}{(1+x)(1+\alpha\,x)}\,dx The first term evaluates to zero. The...
  11. Monoxdifly

    MHB What is the area of the region bounded by the given curves and lines?

    The area of the region y=-x^2+6x, y=x^2-2x, Y-axis, and the line x = 3 is ... A. 16 unit area B. 18 unit area C. \frac{64}{3} unit area D. 64 unit area E. 72 unit area Sorry I couldn't post the graph, but I interpreted it as \int_0^3(-x^2+6x-x^2+2x)dx-\int_0^2(x^2-2x)dx and got \frac{31}{3}...
  12. T

    MHB Understanding Integral Substitution: Finding Equivalent Ranges for Functions

    Hi, I posted a question here a few days ago regarding some questions I've been doing on an online quiz. I seem to be getting stuck on the integral substitution questions. I've been slowly making progress, but some of these questions have been confusing me, and reading up on them is only giving...
  13. Cathr

    Improper integral convergence from 0 to 1

    Homework Statement I have to prove that the improper integral ∫ ln(x)/(1-x) dx on the interval [0,1] is convergent. Homework Equations I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1. The Attempt at a Solution The function can be approximated to ln(x) when it approaches...
  14. lfdahl

    MHB Evaluate the definite integral x/√(e^x+(2+x)^2)

    Evaluate $$I = \int_{-2}^{0} \frac{x}{\sqrt{e^x+(2+x)^2}}\,dx$$
  15. M

    MHB What is the integral of 2^(2x)? tonight, exam is tomorrow

    what is the integral of 2^(2x)? need help tonight, exam is tomorrow teacher says the answer is: 2^(2x) / 2Ln(2) why 2 times Ln(2)?
  16. Math Amateur

    MHB Finite Integral Domains .... Adkins & Weintraub, Proposition 1.5 ....

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as...
  17. Math Amateur

    I Finite Integral Domains .... Adkins & Weintraub, Propn 1.5

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as follows: At the end of the above proof...
  18. C

    Mathematica Cannot do the integral of the Hyper-geometric function?

    Dear friends: It's strange that Mathematica can do the integral of ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##, however, fails when it's changed to ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##. Are there any major differences between this two types? Is it possible to do the second kind of integral...
  19. karush

    MHB 16.1.9 Line Integral over space curves

    Evaluate $\displaystyle \int_C(x+y)ds$ where C is the straight-line segment $x=t, y=(1-t), z=0, $ from (0,1,0) to (1,0,0) ok this is due tuesday but i missed the lecture on it so kinda clueless. i am sure it is a easy one.
  20. karush

    MHB 232.5a Evaluate the double integral

    $\tiny{232.5a}\\ \textsf{Evaluate the double integral}$ \begin{align*}\displaystyle I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\ R&=[0,2]\times[-1,1] \end{align*} Ok, just want to see if I made the first step correct. this looks like simply a rectangle so x and y are basically...
  21. W

    Numerical/Analytical Solution to a Complex Integral

    Homework Statement I have the following integral I wish to solve (preferably analytically): $$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$ where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...
  22. W

    Complex Integral to error function

    Homework Statement I have an integral $$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$ where j and c are complex, which I'd like to write in terms of ## \text{erf}## I'd like to know what would happen to the integral limits as I make the change of variables ##t = jp - c##. 1) As ##p## tends...
  23. karush

    MHB 213.15.4.17 triple integral of bounded by cone and sphere

    $\textsf{Find the volume of the given solid region bounded by the cone}$ $$\displaystyle z=\sqrt{x^2+y^2}$$ $\textsf{and bounded above by the sphere}$ $$\displaystyle x^2+y^2+z^2=128$$ $\textsf{ using triple integrals}$ \begin{align*}\displaystyle V&=\iiint\limits_{R}p(x,y,z) \, dV...
  24. Saurabh

    Solve Hairy Trig Integral: Find Value of 'c

    <Moderator's note: Moved from a technical forum and thus no template.> where a, b, c, d and n, all are positive integers. Find the value of 'c'. ------------------------------- I don't really have a good approach for this one. I just made a substitution u = sinx + cosx I couldn't clear up...
  25. K

    MHB Multivariable calculus line integral work

    calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)
  26. H

    How to solve this integral of an absolute function?

    Homework Statement Homework EquationsThe Attempt at a Solution I think the answer for number 1 , graph somewhat like this I get trouble for 2, 3, etc I (k) = ##\int_{-1}^{1} f(x) dx ## f(x) = ## \mid x^2 - k^2 \mid## 2) k < 1 for negative side ##\int_{-1}^{-k} (x^2 - k^2) dx +...
  27. karush

    MHB 232.15.3.50 Reverse the order of integration in the following integral

    $\textsf{Reverse the order of integration in the following integral }$ \begin{align*}\displaystyle I&=\int_0^1 \int_2^{2e^x}f(x,y) \quad dy \, dx \end{align*} $\textit{From the integral we have that}$ $$0\leq x\leq 1 \quad \textit{and} \quad 2\leq y\leq 2e^x$$ $\textit{So, we get that}$...
  28. Dreadfort

    Integrating a Tricky Cosine Function: Need Help with Substitution

    Homework Statement Solve the integral: [/B] ##\int_{-\infty}^\infty {\frac {\cos(x)}{x^2 + 1}} \, dx##Homework EquationsThe Attempt at a Solution I'm a bit stuck here, so what to substitute for tan \theta so as to compute the integral? Help me out here please
  29. MermaidWonders

    MHB Integral Applications - Hydrostatic Pressure + Force

    Question #1 - A lobster tank in a restaurant is 1 m long by 0.75 m wide by 60 cm deep. Taking the density of water to be 1000 kg/m$^3$, find the water forces (a) on each of the larger sides of the tank; (b) on each of the smaller sides of the tank.
  30. M

    Fredholm Integral Equation Numerically

    Homework Statement A specific problem of the Fredholm integral equation is given as $$\phi(x) = \pi x^2+\int_0^\pi3(0.5\sin(3x)-tx^2)\phi(t)\,dt$$ and the exact solution is ##\phi(x) = \sin 3x##. Homework Equations Nothing comes to mind. The Attempt at a Solution I'm unsure how to approach...
  31. karush

    MHB What is the value of the triple integral for the given limits and function?

    \begin{align}\displaystyle v_{\tiny{s6.15.6.3}}&=\displaystyle \int_{0}^{1}\int_{0}^{z}\int_{0}^{x+z} 6xz \quad \, dy \, dx\, dz \end{align} $\text{ok i kinda got ? with $x+z$ to do the first step?}\\$ $\text{didn't see an example}$
  32. D

    Difficult Vector Field Integral

    <Moderator's note: Image substituted by text.> 1. Homework Statement Given the following vector field, $$ \dfrac{2(x-1)\,dy - 2(y+1)\,dx}{(x-1)^2+(y+1)^2} $$ how do I integrate : The integral over the curve x^4 + y^4 = 1 x^4 + y^4 = 11 x^4 + y^4 = 21 x^4 + y^4 = 31 Homework Equations...
  33. Y

    MHB Trigonometric Integral, weird results

    Hello all, I am trying to solve the integral: \[\int cot(x)\cdot csc^{2}(x)\cdot dx\] If I use a substitution of u=cot(x), I get \[-\frac{1}{2}cot^{2}(x)+C\] which is the correct answer in the book, however, if I do this: \[\int \frac{cos(x)}{sin^{3}(x)}dx\] I get, using a substitution...
  34. MermaidWonders

    MHB Integral Calculus - Spot the Error

    The big blue circle has been put there by my math prof to denote the location of the error in the following solution. Why is this an error? I'm lost. :(
  35. MermaidWonders

    MHB True or False Integral Calculus Question #3

    True or False: If $f(x)$ is a negative function that satisfies $f'(x) > 0$ for $0 \le x \le 1$, then the right hand sums always yield an underestimate of $\int_{0}^{1} (f(x))^2\,dx$. - - - Updated - - - Would it be true since right hand Riemann sums for a negative, increasing function will...
  36. MermaidWonders

    MHB True or False Integral Calculus Question #2

    True or False: Let $F(x)$ be an antiderivative of a function $f(x)$. Then, $F(2x)$ is an antiderivative of the function $f(2x)$.
  37. MermaidWonders

    MHB True or False Integral Calculus Question #1

    True or False: If $$h(t) > 0$$ for $$0 \le t\le 1$$, then the function $$H(x) = \int_{0}^{x} h(t)\,dt$$ is concave up for $$0 \le t\le 1$$.
  38. karush

    MHB How do you evaluate the spherical coordinate integral at 244.15.7.24?

    $\tiny{244 .15.7.24}$ $\textsf{Evaluate the spherical coordinate integral}\\ \begin{align}\displaystyle DV_{24}&=\int_{0}^{3\pi/4} \int_{0}^{\pi} \int_{0}^{1} \, 5\rho^3 \sin^3 \phi \, d\rho \, d\phi \, d\theta \\ &=\int_{4}^{3\pi/4}...
  39. hugo_faurand

    B Calculate the expression of the antiderivative

    Hello everyone ! I've started to work on integral and I wonder if it's possible to calculate the expression of the antiderivative with the expression of the "integrand"1 rather than use a table with the function and its antiderivative. Thank you in advance ! 1( I'm french and I d'ont know the...
  40. K

    Rearranging an equation involving an integral

    Homework Statement Homework Equations How do we get y(x) ? should y'(x) in the sqrt not cancel the y/(x) on the RHS ? Does y(x) comes back because of integrating 1 with dy ? The Attempt at a Solution w(x)y'(x) = A (1 +y'(x)2)1/2 (w(x)y'(x) )2 = ( A (1 +y'(x)2)1/2 )2 w(x)2y'(x)2 = A2...
  41. starstruck_

    Integral (Trapezoidal rule and mid point rule)

    Homework Statement find: ∫13e^(1/x) upper bound: 2 lower bound: 1 using the trapezoidal rule and midpoint rules estimate the errors in approximation Homework Equations I've done the approximations using the trapezoidal rule and midpoint rule, but I can't figure out how to calculate...
  42. D

    I Show that the integral of the Dirac delta function is equal to 1

    Hi, I am reading the Quantum Mechanics, 2nd edition by Bransden and Joachain. On page 777, the book gives an example of Dirac delta function. $\delta_\epsilon (x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$ I am wondering how I can show $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$...
  43. N

    I Solving Integral for All n≥2 | Evans PDE's (Page 48)

    In the book from Evans on PDE's (page 48) I came across this integral. Here r > 0 and \delta is an arbitrarily small number. Could you give me some hint on how to solve this integral for all integers n\geq2 , i.e why does it go to zero as t approaches zero from the right side.
  44. aphirst

    I Derivative and Parameterisation of a Contour Integral

    As part of the work I'm doing, I'm evaluating a contour integral: $$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$ along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...
  45. J

    Integral Neutron Flux: Getting Results with MCNP - Juan Galicia-Aragon

    Hello everyone I am trying to obtain the integral neutron flux based on the results obtained with MCNP (neutron spectrum calculation) for each energy bin (51 neutron energy bins). I have seen in many papers the calculation of the differential neutron flux multiplying the neutron flux results of...
  46. Z

    MHB How to calculate this type of integral, Thanks

    $$\int {z}^{2}\arcsin\left({\frac{a+\sqrt{392-{a}^{2}-2{z}^{2}}}{2 \sqrt{196-{z}^{2}}}}\right) dz$$ $$\int {z}^{2}\arcsin\left({\frac{a}{\sqrt{196-{z}^{2}}}}\right) dz$$
  47. Z

    MHB How to calculate this type of integral

    Could anyone can tell me how to calculate this type of intergretion. Thanks very much $$\int\frac{{y}^{3}}{(196 - {y}^{2})\times \sqrt{196 - {y}^{2} - {a + y}^{2}}}$$
  48. D

    Meaning of the FFT of a Poynting Vector integral, reflection coefficient

    Hello, For calculating the mean power at a specific cross section of a waveguide, one can calculate the mean value of the temporal function of Poynting Vector, P(t), where P(t) is the ExHy-EyHx. Note that I am not talking about phasors or a sinusoidal state. If I integrate over the waveguide...
  49. SemM

    I Can you simplify this integral or do you need more background knowledge?

    Hi, I have the following integral which I am not confident on how to interpret (solve):\begin{equation} \alpha \bigg( \int_0^L [\frac{d^3}{dx^3}\phi] \psi dx - \int_0^L [\frac{d^3}{dx^3}\psi] \phi dx \bigg) \end{equation} at this stage, I am not sure which rule to use to solve each of the two...
  50. J

    A Maximization Problem: Double Int. w/ C not Dependent on Integrals

    Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2} $$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...
Back
Top