Definite integral Definition and 391 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. chwala

    I The net area and total area involving definite integrals

    on checking, i am informed that there is net area and total area. I would like more clarity on this. In other words is calculating the definite integral equivalent to calculating net area? as opposed to calculating the area under the curve (total area)? how do we distinguish the two? Thanks
  2. A

    Electromagnetism: Force on a parabolic wire in uniform magnetic field

    I know the easier method/trick to solve this which doesn't require integration. Since parabola is symmetric about x-axis and direction of current flow is opposite, vertical components of force are cancelled and a net effective length of AB may be considered then ##F=2(4)(L_{AB})=32\hat i## I...
  3. Nitacii

    Integrate source terms for test EM field in Kerr spacetime

    Hello, the Homework Statement is quite long, since it includes a lot of equations so I will rather post the as images as to prevent mistypes. We need to find the integral where with $$ J_m =(\sqrt{2}(r−ia\cos⁡θ))^{−1} i(r^2+a^2)\sin⁡(θ)j, $$ $$ J_n = - \frac{a \Delta}{ 2 \Sigma} \sin(\theta...
  4. S

    I Result of this integral in large Lambda limit

    $$\int \text{d}^4 q \, \frac{1}{(q^2 + m^2)\left(1+\frac{q^2}{\Lambda^2}\right)^2} =2 \pi^2\left(\frac{\Lambda^2}{2}-m^2 \log \frac{\Lambda}{m} \right) + o(\Lambda^0)$$ How to get this result? The notation ##o(\Lambda^0)## means all terms constant in Lambda, which we ignore because we are...
  5. S

    Solving this definite integral using integration by parts

    Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks
  6. chwala

    Find the value of the definite integral

    Looking at integration today...i will go slow as i also try finish other errands anyway; i am thinking along these lines; $$\int \sqrt{(ax^2+bx+c)} dx=\sqrt{a}\int \sqrt{\left[x+\frac{b}{2a}\right]^2+\left[\frac{4ac-b^2}{4a^2}\right]} dx$$ ... Therefore, $$\int_0^2 \sqrt{(8t^2+16t+16)}...
  7. crememars

    Finding a definite integral from the Riemann sum

    Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...
  8. crememars

    Identifying variables from Riemann sum limits

    Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
  9. mcastillo356

    I Express the limit as a definite integral

    Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt: (i) General Riemann Sums Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
  10. Euge

    POTW Definite Integral of a Rational Function

    Evaluate the definite integral $$\int_0^\infty \frac{x^2 + 1}{x^4 + 1}\, dx$$
  11. chwala

    Find the value of the definite integral

    Find question here, My approach, using cosine sum and product concept, we shall have; ##\cos (A+B)-\cos (A-B)=-2\sin A\sin B## ##⇒\cos D-\cos C=-2\sin\dfrac{C+D}{2} \sin\dfrac {C-D}{-2}## ##⇒-3[\cos(A+B)-\cos(A-B)]=6\sin A sinB## We are given ##A=4θ## and ##B=2θ##, therefore, ##⇒-3[\cos...
  12. chwala

    Evaluate the definite integral in the given problem

    My interest is on the highlighted part only. Find the problem and solution here. This is clear to me (easy )...i am seeking an alternative way of integrating this...or can we say that integration by parts is the most straightforward way? The key on solving this using integration by parts...
  13. A

    A How to solve this definite integral?

    I would like to solve the integral underneath: $$\displaystyle \int_{0}^{x}\!-{\frac {\lambda\,{{\rm e}^{-\lambda\,t}}{\beta}^{\alpha} \left( -\lambda+\beta \right) ^{-\alpha} \left( -\Gamma \left( \alpha \right) +\Gamma \left( \alpha, \left( -\lambda+\beta \right) t \right) \right)}{\Gamma...
  14. DaalChawal

    MHB How Large Can This Integral Be?

  15. MountEvariste

    MHB Definite integral involving sine and hyperbolic sine

    Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$
  16. anemone

    MHB Prove: Integral Inequality for Convex Function

    Let $f:[1,\,13]\rightarrow R$ be a convex and integrable function. Prove that $\displaystyle \int_1^3 f(x)dx+\int_{11}^{13} f(x)dx\ge \int_5^9 f(x)dx$,
  17. chwala

    Find the value of this definite integral

    ##\int_0^5 [-x^3+3x^2+6x-8\,]dx## ##\int_0^1 [-x^3+3x^2+6x-8\,]dx= |-\frac {17}{4}|## ##\int_1^4 [ -x^3+3x^2+6x-8\,]dx= 16## ##\int_4^5[-x^3+3x^2+6x-8\,]dx= |-\frac {49}{4}|## Therefore, total area is ##|-\frac {17}{4}|+ 16+|-\frac {49}{4}|=32.5## now where my problem is,... my colleague...
  18. caffeinemachine

    MHB Definite integral involving a lot of exponentials.

    Problem: Evaluate $$ \int_{0}^\infty \frac{e^{3x} - e^x}{x(e^x + 1)(e^{3x} + 1)}\ dx $$ Attempt. I substituted $y=e^x$, thus $dx = dy/y$, which turns the above integral to $$ \int_{1}^\infty \frac{y^2 - 1}{(\log y)(y+1)(y^3+1)}\ dy = \int_{1}^\infty \frac{y-1}{(\log y) (y^3+1)} \ dy $$ I am...
  19. S

    Mathematica Definite integral with some unknown variables

    I am trying to evaluate an integral with unknown variables ##a, b, c## in Mathematica, but I am not sure why it takes so long for it to give an output, so I just decided to cancel the running. The integral is given by, ##\int_0^1 dy \frac{ y^2 (1 - b^3 y^3)^{1/2} }{ (1 - a^4 c^2 y^4)^{1/2} }##
  20. A

    I Derivative of a definite integral

    If $$F(x)=\int_{a}^{b}f(x)dx$$ implies $$F'(x)=\int_{a}^{b}f'(x)dx$$?
  21. Saracen Rue

    I Find the value of this definite integral in terms of t, s and alpha

    One of the maths groups I'm apart of on Facebook posts (usually) daily maths challenges. Typically they act as small brain teaser for when I wake up and I can solve them without much trouble. However, today's was more challenging: (Note: blue indicates a variable and red indicates a constant)...
  22. archaic

    Trigonometric definite integral of 1/(4-sqrt(x))

    This could be solved by the substitution ##u=\sqrt x##, but I wanted to do it using a trigonometric one. The answer is false, but I don't see the wrong step. Thank you for your time! [Poster has been reminded to learn to post their work using LaTeX]
  23. lfdahl

    MHB How Can the Given Definite Integral Identity Be Proven for Any Natural Number n?

    Show, that the identity \[\int_{0}^{1}\frac{x^{n-1}+x^{n-\frac{1}{2}}-2x^{2n-1}}{1-x}dx = 2\ln2\] - holds for any natural number $n$.
  24. M

    Expressing a limit as a definite integral

    Express the limit ##lim_{n\rightarrow\infty} \sum_{i=1}^n \frac2n\ (1+\frac {2i-1}{n})^\frac13## This is worked example but I would like to ask about the points I don't understand in the book. "We want to intepret the sum as a Riemann sum for ##f(x)=(1+x)^3## The factor ##\frac2n## suggests...
  25. Saracen Rue

    B Is the Definite Integral of the Function Exactly 1?

    Hi everyone, I am trying to find the definite integral of a function (see attached image) from 0 to infinity using Wolfram alpha. I'm just looking for some verification on if the integral actual is equal to exactly 1, or if there's some rounding errors going on. Thank you for your time :)
  26. J

    Definite Integral of Product/Composite Function Given Graph

    Homework Statement Given the graph of f(x) shown below, find the value of the integral. Photo attached. Homework Equations [/B] ∫23 5x·f(x2)dx The Attempt at a Solution [/B] I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x2))...
  27. T

    MHB Definite integral with Riemann sums

    Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...
  28. T

    MHB Basic definite integral question

    Hey, I've got this problem I've been doing, but I'm not sure if my approach is right. My textbook has pretty much less than a paragraph on this sort of stuff. My thinking was that since an integral is a sum, in order to get the range from 0 to 8, we should just be able to add or subtract the...
  29. lfdahl

    MHB Is the definite integral ∫ [arcsin(1/x)-1/x]of indeterminate form?

    Is the definite integral $$\int_{1}^{\infty}\left(\arcsin \left(\frac{1}{x}\right)-\frac{1}{x} \right)\,dx$$ of indeterminate form or not? Prove your statement.
  30. Krushnaraj Pandya

    Definite integral using only properties

    Homework Statement evaluate the following definite integral with limits 0 to 1## ∫log(sin(πx/2)) dx ## 2. The attempt at a solution I used ##∫f(x) = ∫f(a+b-x)## to get ## I=∫log(cos(πx/2))## with the same limits. Adding them and using ##log(m)+log(n)=log(mn)## and ##2sinxcosx=sin2x## I got...
  31. ubergewehr273

    Problem involving a definite integral

    Homework Statement Refer the image. Homework Equations Integral (e^x)dx from 0 to 1=e-1.The Attempt at a Solution Refer the other image. The graph of e^(x^2) increases more slowly than e^x till x=1. So 'a' is clearly greater than 1. Is this right?
  32. 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...
  33. lfdahl

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

    Evaluate $$I = \int_{-2}^{0} \frac{x}{\sqrt{e^x+(2+x)^2}}\,dx$$
  34. 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...
  35. H

    Find f(x) which satisfies this integral function

    Homework Statement find f(x) which satisfies f(x) = x + ##\frac{1}{\pi}## ##\int_{0}^{\pi} f(t) \sin^2{t} \ d(t)## Homework EquationsThe Attempt at a Solution to solve f(x), I have to solve the integral which contains f(t). And f(t) is the f(x) with variable t? if yes, I will get integral...
  36. lfdahl

    MHB Definite integral challenge ∫ln(2−2cosx)dx=0

    Prove, that the definite integral $$\int_{0}^{\pi}\ln (2-2\cos x)dx = 0.$$
  37. T

    MHB Definite integral of square root+cube root

    Dear all, Please solve this integral: I tried integral by substitution, but failed. Wolframalpha shows the result is 6, but I don't know how to proceed it. Can it be solved by elementary function?
  38. W

    I Integrating scaled and translated indicator function

    I am struggling to evaluate the following, relatively easy, integral (it might be because its early on a monday morning): $$I_{jk}(a)=\int_0^a\chi_{[0,1)}(2^jx-k)\,dx,$$ where ##\chi_{[0,1)}(x)## denotes the indicator function on ##[0,1)## and ##j,k## are both integers. My idea is to rewrite the...
  39. MarkFL

    MHB Definite Integral ∫xe^(ax)cos(x)dx

    Evaluate the following: I=\int_0^{\infty} xe^{ax}\cos(x)\,dx where $a<0$
  40. lfdahl

    MHB Definite integral ∫(cos4x−cos4α)/(cosx−cosα)dx

    Evaluate the definite integral:\[I = \int_{0}^{\pi}\frac{\cos 4x - \cos 4\alpha }{\cos x - \cos \alpha }dx\]- for some $\alpha \in \mathbb{R}.$
  41. P

    Primitive of a definite integral

    Homework Statement I need find the function ##F(x)## . Homework Equations ##\int_0^r F(x)dx = \frac{r^3}{(r^2+A)^{3/2}}+N## where ##A,N## are constants. The Attempt at a Solution I tried using some function of test, for instance the derivative of the right function evaluated in x. But , i...
  42. lfdahl

    MHB Definite integral challenge ∫cos2017xsin2017xdx

    Calculate the following definite trigonometric integral: \[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\].
  43. J

    I Taylor expansions and integration.

    I have a short doubt: Let f(x) be a fuction that can't be integrated in an analytical way . Is anything wrong if I expand it in a Taylor' series around a point and use this expansion to get the value of the definite integral of the function around that point? Suppose that the interval between...
  44. doktorwho

    How to Prove the Integral Property for Definite Integrals

    Homework Statement Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving:: Homework Equations 3. The Attempt at a Solution [/B] As no...
  45. Mr Davis 97

    I Solving a definite integral by differentiation under the integral

    Say we have the following integral: ##\displaystyle \int_0^1 \frac{\log (x+1)}{x^2+1}##. I know how to do this integral with a tangent substitution. However, I saw another method, which was by differentiating ##f## under the integral with respect to the parameter ##t##, where we let...
  46. D

    Determine the truth of the following statements

    Homework Statement ##f(x) = \begin{cases} -\frac{1}{1+x^2}, & x \in (-\infty,1) \\ x, & x \in [1,5]\setminus {3} \\ 100, & x=3 \\ \log_{1/2} {(x-5)} , & x \in (5, +\infty) \end{cases}## For a given function determine the truth of the folowing statements and give a brief explanation: a) Function...
  47. 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 =...
  48. 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...
  49. 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...
  50. 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 =...
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