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. ShayanJ

    A Why can't I use contour integration for this integral?

    Consider the integral ##\displaystyle \int_{-\infty}^\infty \frac{e^{-|x|}}{1+x^2}dx ##. I should be able to use contour integration to solve it because it vanishes faster than ## \frac 1 x ## in the limit ## x \to \infty ## in the upper half plane. It has two poles at i and -i. If I use a...
  2. kaniello

    I Calculating the Convolution Integral for General Math Community

    Dear "General Math" Community, my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right...
  3. S

    How to parameterize these surfaces?

    Homework Statement Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0## Homework Equations Parameterizations: ##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le...
  4. F

    Double Integral: Error in Calculation?

    Homework Statement I have calculate my double integral using wolfram alpha , but i get the ans = 312.5 , but according to the book , the ans is = 0 , which part of my working is wrong Homework EquationsThe Attempt at a Solution Or is it z =0 , ? i have tried z = 0 , but still didnt get the...
  5. williamwong0402

    How to determine this integral? Thank you

    Homework Statement Homework Equations k∫[ƒ(x)]n ƒ'(x) dx The Attempt at a Solution i tried to using algebraic substitution to determine that i had let u = 1-x or X2-2x+1 or x or root(x) but it still cannot solve it. Please give me hint how to solve it. Thank you [/B]
  6. T

    MHB Integral that converges or diverges?

    I have: $$\int_{1}^{3} \frac{1}{\sqrt{3 - x}} \,dx$$ I can do $u = \sqrt{3 -x}$, and $du = \frac{1}{2 \sqrt{3 - x}} dx $ and $dx = 2 \sqrt{3 - x} du $. Plug into original equation: $$\int_{1}^{3} \frac{2 u }{u} \,du$$ and $2 \int_{1}^{3} \,du = 2u = 2 \sqrt{3 - x} + C$ So $(2\sqrt{3 - 3})...
  7. T

    MHB Divergent or Convergent Integral

    I have: $$\int_{1}^{2} \frac{1}{x lnx} \,dx$$ I can set $u = lnx$, therefore $du = \frac{1}{x} dx$ and $xdu = dx$. Plug that into the original equation: $$\int_{1}^{2} \frac{x}{x u} \,du$$ Or $$\int_{1}^{2} \frac{1}{ u} \,du$$ Therefore: $ln |u | + C$ and $ln |lnx | + C$ So I need to...
  8. ShayanJ

    A Ground state wave function from Euclidean path integral

    From the path integral approach, we know that ## \displaystyle \langle x,t|x_i,0\rangle \propto \int_{\xi(0)=x_i}^{\xi(t_f)=x} D\xi(t) \ e^{iS[\xi]}##. Now, using ## |x,t\rangle=e^{-iHt}|x,0\rangle ##, ## |y\rangle\equiv |y,0\rangle ## and ## \sum_b |\phi_b\rangle\langle \phi_b|=1 ## where ## \{...
  9. J

    I Can the Complex Integral Problem Be Solved Using Residue Theorem?

    I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...
  10. lep11

    Calculate the following line integral

    Homework Statement Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##. Homework EquationsThe Attempt at a Solution The answer is 0. Here's my work. However, this method requires that you are...
  11. F

    Correcting Errors in Conservative Line Integral Calculation

    Homework Statement I am having question with part c , for both c1 and c2 , here's my working for c1 , i didnt get the ans though . My ans is -5 , but the given ans for c1 and c2 is 27 , is the ans wrong ? Or which part i did wrongly ? Homework EquationsThe Attempt at a Solution
  12. G

    A A problem about branch cut in contour integral

    Hello. I have a difficulty to understand the branch cut introduced to solve this integral. \int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]} here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
  13. T

    MHB Determining if integral converges or diverges

    $\int_{3}^{\infty} \frac{1}{\sqrt{x} - 1} \,dx$ I need to find if this converges or diverges. I'm trying u-substitution, so $u = \sqrt{x} - 1$. Therefore, $du = \frac{1}{2\sqrt{x}} dx$. I'm not sure how to proceed from here.
  14. dykuma

    Contour integral using residue theorem

    Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...
  15. maxhersch

    Estimate Vector Field Surface Integral

    I assume this is a simple summation of the normal components of the vector fields at the given points multiplied by dA which in this case would be 1/4. This is not being accepted as the correct answer. Not sure where I am going wrong. My textbook doesn't discuss estimating surface integrals...
  16. J

    MHB Evaluating Definite Integral $I$

    Evaluation of $\displaystyle \int^{\frac{1}{2}}_{0}\frac{1}{(1-2x^2)\sqrt{1-x}}dx$ $\bf{Try::}$ Let $\displaystyle I = \int^{\frac{1}{2}}_{0}\frac{1}{(1-2x^2)\sqrt{1-x}}dx$ Put $1-x=t^2\;,$ Then $dx=-2tdt$ So $\displaystyle I = \int^{1}_{\frac{1}{2}}\frac{2t}{\left[1-2(1-t^2)^2\right]t}dt =...
  17. Sunny29

    A How to Solve an Integral Equation Involving Exponential Functions?

    Please anyone can help solve this integral equation e^t+e^t ∫ (t, 0 ) e^(-τ) x f(τ) dτ
  18. S

    B What is the integral of (x^2+x)/(2x+1) when split into manageable pieces?

    I know integral of (2x+1)/(x^2+x). but i don't know integral of (x^2+x)/(2x+1). I'm very curious... please answer me...
  19. garylau

    Where is pi/4 coming from in the line integral?

    Sorry where is pi/4 coming from in the line integral(section 3)? because i think it should be 1/2=tan(theta) which theta is 26.5651... it is impossible that the angle is pi/4? where is pi/4 coming from inside the circle? thank
  20. Cjosh

    Calculating the definite integral using FTC pt 2

    Homework Statement Sorry that I am not up on latex yet, but will describe the problem the best I can. On the interval of a=1 to b= 4 for X. ∫√5/√x. Homework EquationsThe Attempt at a Solution My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...
  21. MyNameIsNicholas

    How do you "read" this formula?

    Homework Statement No actual work, could just use some assistance in understanding formulas involving the centroid of an object, specifically with integrals. For example, how would you understand the following formula(s) (as seen in part 2)? I understand that the centroid is the sum of all the...
  22. T

    MHB Integral using trig substitution

    I have $$\int_{}^{} \frac{1}{\sqrt{1 - x^2}} \,dx$$ I can let $x = \sin\left({\theta}\right)$ then $dx = cos(\theta) d\theta$ then: $$\int_{}^{} \frac{cos(\theta) d\theta}{\sqrt{1 - (\sin\left({\theta}\right))^2}}$$ Using the trig identity $1 - sin^2\theta = cos^2\theta$, I can simplify...
  23. toforfiltum

    Evaluating Cartesian integral in polar coordinates

    Homework Statement Transform given integral in Cartesian coordinates to one in polar coordinates and evaluate polar integral. ##\int_{0}^3 \int_{0}^x \frac {dydx}{\sqrt(x^2+y^2)}## Homework EquationsThe Attempt at a Solution I drew out the region in the ##xy## plane and I know that ##0 \leq...
  24. core1985

    Finding the Right Normalization Constant for Gaussian Integrals

    Hello I have tried gaussian integrals does gaussian integrals have this general form formula? if not then weather i do integration by parts or what just needed a hint to solve it correctly
  25. Leechie

    Evaluating complex integral problem

    Homework Statement I'm having some trouble evaluating the integral $$\int^\infty_{-\infty} \frac{\sqrt{2a}}{\sqrt{\pi}}e^{-2ax^2}e^{-ikx}dx$$ Where a and k are positive constants Homework Equations I've been given the following integral results which may be of help $$\int^\infty_{-\infty}...
  26. T

    Work problem - Rope, pulley and brick (applied integration)

    If a brick is pulled across the floor by a rope thruogh a pulley, 1 meter above the ground - and work = W, where W = 10N , (in Newton).Show that the horizontal component of W, which is pulling the brick has the size \frac{10x}{\sqrt{1+x^2}} (*) Use this to calculate the amount of work needed...
  27. A

    I Integral with complex oscillating phase

    Does there exist and analytical expression for the following integral? I\left(s,m_{1},m_{2},L\right)=\sum_{\boldsymbol{n}\in\mathbb{N}^{3}\backslash\left\{ \boldsymbol{0}\right\}...
  28. X

    Find the multiple integral for a modeled problem

    Homework Statement A 12 foot light pole stands at the corner of an 8 foot by 10 foot rectangular picnic blanket spread out on the ground. A bee flies in a straight line from a point P on the pole to a point Q on the blanket. Set up a multiple integral whose value represents the average length...
  29. A

    Potential difference due to a continuous charge distribution

    This is my first time using this site so please excuse me if I missed any guidelines. 1. Homework Statement A plastic rod having a uniformly distributed charge Q=-25.6pC has been bent into a circular arc of radius R=3.71cm and central angle ∅=120°. With V=0 at infinity, what is the electric...
  30. S

    I Gaussian integral in two dimensions

    I am trying to evaluate the following integral. ##\displaystyle{\int_{-\infty}^{\infty}f(x,y)\ \exp(-(x^{2}+y^{2})/2\alpha)}\ dx\ dy=1## How do you do the integral above?
  31. toforfiltum

    Integrating triple integral over region W

    Homework Statement $$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##. Homework EquationsThe Attempt at a Solution Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have...
  32. F

    Integral x/(1-x) via power series?

    So, ∫x/(1-x)... can I solve this as a power series ∫(x*Σ x^n) = ∫(Σ x^(n+1))= (1/(n+2)*Σ x^(n+2))? Is this correct? I know there is other ways to do it... But should this be correct on a test? This solution is more fun..
  33. D

    I Integral of f(x)*x: Is It Possible?

    Is it possible to do an integral of f(x)*x without knowing f(x)?
  34. terryds

    Definite integral as Riemann sums

    Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...
  35. K

    I Integration Method Used to Transform Equation 1 into Equation 2?

    Hi, what kind if integration used on equation 1 so it turned into equation 2? this does not look like integration by parts. and where (x-x0) appeared from instead of (k-k0) ? thanks for your help.
  36. Y

    MHB Calculus - definite integral

    Consider the definite integral ∫202x(4−x2)1/5 dx. What is the substitution to use? u= 4-x^2 Preview Change entry mode (There can be more than one valid substitution; give the one that is the most efficient.) For this correct choice, du/dx= -2x Preview Change entry mode If we make this...
  37. pac0master

    Calculating Half life Decay in an Open system

    [Question] So I was thinking about Physics for some time and for the sake of curiosity I've came with this problem: Let's say we have a liquid flowing into a system with infinite space. The flow is constant ( F ) The liquid decays over time with a half life ( λ ) We're looking for the Total...
  38. O

    Integral of Sqrt(x)*e^-x: Step-by-Step Solution

    Homework Statement Evaluate the following integral: ∫0∞ √(x)* e-x dx Homework Equations ∫0∞ e-x2 dx = (√π)/2 The Attempt at a Solution So far this is what I've done: u = x1/2 du = 1/2 x-1/2 2 ∫ e-u2 u2 du Now, I'm not really sure what to do? Or if what I've done so far is leading me down...
  39. T

    I Derivative of A Def. Integral Equals Another Def. Integral?

    I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...
  40. N

    MHB Evaluation of definite integral (is it correct?)

    Is this question correct? We are given to evaluate: \int_0^2 \left(e^x-e^{-x}\right)^2\,dx 2\left(\frac{1}{2}\sinh(x)-x\right) 2\left(\frac{1}{2}\sinh(2\cdot2)-2\right)-2\left(\frac{1}{2}\sinh(2\cdot0)-0\right) \sinh(4)-4
  41. C

    MHB What is the correct solution for $\int_{0}^{1}x\sqrt{18-2x^2} \,dx$?

    Hello, I need some help finding the integral $\int_{0}^{1}x\sqrt{18-2x^2} \,dx$ Let $u= 18-2x^2$ $du=-4xdx$ $-1/4 \int_{16}^{18} \sqrt{u} \,du$= $36 sqrt(2) -128/3$ I am getting the wrong solution: $9\sqrt(2)-32/3$
  42. S

    Helicity integral in differential forms

    Homework Statement Let ##V^{3}(t)## be a compact region moving with the fluid. Assume that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in...
  43. O

    Double integral, find volume of solid

    Homework Statement Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders: y = 1 − x2, y = x2 − 1 and the planes: x + y + z = 2 4x + 5y − z + 20 = 0 Homework Equations ∫∫f(x,y) dA The Attempt at a Solution So I solved for z in the plane...
  44. N

    MHB Finding area (application of definite integral)

    Hi, I am stuck on this question and was wondering if anyone could help me. The topic is integral equations. A block of land is bounded by two fences running North-South 5 km apart a fence line which is approximated by the function N=0.5E and a road which is approximated by the curve...
  45. C

    Double integral to compute capacitance

    Hi everyone. I have quite a basic doubt, and I thought you could help me. Consider the figure: The cylinders S1 is held at a constant potential, and the same applies for the ring identified by S0. All the surroundings are filled with an insulator material. I want to calculate che capacitance...
  46. D

    Issue with Double Integral

    Homework Statement Find the volume of the given solid: Under the surface z = xy and above the triangle with verticies (1,1), (4,1) and (1,.2) Homework Equations Double Integral The Attempt at a Solution I drew the triangle, and found the the equations of the lines to be: x = 1; y = 1; y = -3x...
  47. lep11

    Expressing defined integral as composition of differentiable functions

    Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework EquationsThe Attempt at a Solution Honestly, I have...
  48. K

    Integral of a hyperbolic function

    Homework Statement $$\int \tanh=?$$ Homework Equations $$\cosh^2-\sinh^2=1$$ $$(\tanh)'={\rm sech}^2=\frac{1}{\cosh^2},~~(\coth)'=-{\rm csch}^2=-\frac{1}{\sinh^2}$$ $$({\rm sech})'=\left( \frac{1}{\cosh} \right)'=-{\rm sech}\cdot\tanh=-\frac{\sinh}{{\rm cosh}^2}$$ $$({\rm csch})'=\left(...
  49. Genilson

    I Limit with integral and absolute value

    Hello good evening to all, I was studying here and got stuck with this. I solved the integral and got [x+sin(x) -1] and that´s the farthest that I got. I would appreciate the help.
  50. R

    Surface Integral (Integral Setup)

    Homework Statement I'm just required to setup the integral for the question posted below Homework EquationsThe Attempt at a Solution So solving for phi @ the intersection of the sphere and the plane z=2: z = pcos(phi) 2 = 3cos(phi) phi = arccos(2/3) so my limits for phi would go from 0 to...
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