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. 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 $\alpha \in \mathbb{R}.$
  2. W

    Solve Integral Equation: xe-axcos(x)dx from 0 to ∞

    Homework Statement Solve from x = 0 to x = ∞, ∫xe-axcos(x)dx Homework EquationsThe Attempt at a Solution I have a solution for the integral ∫e-axcos(x)dx at the same limits, and I feel that the result might be of use, but have no idea how to manipulate the integral above such that I can use...
  3. Y

    MHB Differentiating wrt constant to evaluate integral

    What is the justification for differentiating some integrals with respect to constants in order to obtain result, i.e. ∂/∂a[∫e^(− ax^2).dx] =∫-x^2.e^(-ax^2) dx?I mean what if we say "a" was 3 then differentiating wrt 3 would have no significance?How can we treat it like a multivariable function :/
  4. karush

    MHB 232.15.4.46 Change the Integral order then evaluate

    $\tiny{232.15.4.46}$ $\textsf{Change the order then evaluate}$ \begin{align*}\displaystyle I&=\int_{0}^{1}\int_{0}^{2}\int_{2y}^{4} \frac{5\cos(x^2)}{2z} \, dx \, dy \, dz \end{align*} ok I presume the change that should be made is... altho I don't know what represents x or y...
  5. ZARATHUSTRA

    I Is a Line Integral Zero if the Vector Field is Not Conservative?

    calculate the line integral for a vector field F= -xy⋅j over a circle which is c(t)=costi+sintj, so I used x=cost y=sint and ∫(0 to 2pi) -(sintcost)(cost)dt=(cos^3(2pi)-cos^3(o))/3=0 now here is the problem, if this enclosed line integral is zero then why is the vector field not conservative?
  6. P

    Show this integral defines a scalar product.

    Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...
  7. karush

    MHB 15.3.65 Rewriting double integral to infnty

    \begin{align*}\displaystyle \int_{\alpha}^{\beta}\int_{a}^{\infty} g(r,\theta) \, rdr\theta =\lim_{b \to \infty} \int_{\alpha}^{\beta}\int_{a}^{b}g(r,\theta)rdrd\theta \end{align*} $\textit{Evaluate the Given}$ \begin{align*}\displaystyle &=\iint\limits_{R} e^{-x^2-y^2} \, dA \\ (r,\theta) \, 2...
  8. AutumnWater

    I Q about finding area with double/volume with triple integral

    So when finding the Area from a double integral; or Volume from a triple integral: If the curve/surface has a negative region: (for areas, under the x axis), (for volumes, below z = 0 where z is negative) What circumstances allow the negative regions to be taken into account as positive when...
  9. lfdahl

    MHB Double integral challenge

    Evaluate the double integral: \[I = \int \int _R\frac{1}{(1+x^2)y}dxdy\] - where $R$ is the region in the upper half plane between the two curves: $2x^4+y^4+ y = 2$ and $x^4 + 8y^4+y = 1$.
  10. karush

    MHB Evaluating Improper Integrals in Polar Coordinates

    15.3.65 Improper integral arise in polar coordinates $\textsf{Improper integral arise in polar coordinates when the radial coordinate r becomes arbitrarily large.}$ $\textsf{Under certain conditions, these integrals are treated in the usual way shown below.}$ \begin{align*}\displaystyle...
  11. karush

    MHB 15.3.50 Double integral of circle and graph

    $\displaystyle \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \sqrt{x^2+y^2} \, dydx=\frac{\pi}{6}$ this was the W|A answer but how ? also supposed to graph this but didn't know the input for desmos
  12. 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...
  13. karush

    MHB 232.q1.2c Double integral with absolute value in integrand

    $\displaystyle \int_{-1}^{1} \int_{-2}^{3}(1-|x|) \,dy\,dx$ ok i was ? about the abs
  14. C

    I Huygens principle from Kirchoff Integral (sign doubt)

    Consider the Kirkoff integral theorem and the Huygens -Fresnel principle/formula (both from Wikipedia): KIT The Kirchoff integral for monochromatic wave is: $$U({\mathbf {r}})={\frac {1}{4\pi }}\int _{S'}\left[U{\frac {\partial }{\partial {\hat {{\mathbf {n}}}}}}\left({\frac...
  15. karush

    MHB 232.q1.5e Double trig. integral

    $$\iint_\limits{R}(3x^5-y^2\sin{y}+5) \,dA$$ $$R=[(x,y)|x^2+y^2 \le 5]$$
  16. karush

    MHB Double Integral 232.q1.5a Calculation

    $\tiny{232.q1.5,a}$ \begin{align*}\displaystyle I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\ R&=[0,2]\times[-1,1] \end{align*} would this be $$\int_{-1}^{1} \int_{0}^{2}xy\sqrt{x^2+y^2} \,dx \, \, dy $$
  17. RJLiberator

    Isothermal Work Calculation for an Imperfect Gas

    Homework Statement An imperfect gas obeys the equation (p+\frac{a}{V^2_m})(V_m-b)=RT where a = 8*10^(-4)Nm^4mol^(-2) and b=3*10^(-5)m^3mol^(-1). Calculate the work required to compress 0.3 mol of this gas isothermally from a volume of 5*10^(-3)m^3 to 2*10^(-5)m^3 at 300K. Homework Equations...
  18. D

    Trying to find this double integral using polar coordinates

    Homework Statement question : find the value of \iint_D \frac{x}{(x^2 + y^2)}dxdy domain : 0≤x≤1,x2≤y≤x Homework Equations The Attempt at a Solution so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x and i decided to convert...
  19. karush

    MHB How do we evaluate the double integral with $15x^2$ inside and all the R values?

    ok just seeing if I have this set up correctly before evaluate.. where does $15x^2$ come from? if $15x^2$ is inside this why would we need all the R values
  20. lfdahl

    MHB Evaluate the integral ∫[arctan(ax)−arctan(bx)]/xdx

    Evaluate the integral: \[ \int_{0}^{\infty}\frac{\arctan(ax)-\arctan(bx)}{x}dx\] where $a,b \in \mathbb{R}_+$
  21. karush

    MHB 15.2.87 Write the following integrals as a single iterated integral.

    Draw the regions of integration and write the following integrals as a single iterated integral. $$\displaystyle\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy + \int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$$ ok haven't done this before so kinda clueless
  22. karush

    MHB 15.2.78 But it asks for a double integral

    Use double integral to compute the area of the region bounded by $y=4+4\sin{x}$ and $y=4-4\sin{x}$ on the interval $\left[0,\pi\right]$ ok it looks easier to do this in one $\int$ but it asks for a double $\int\int$ so ?
  23. karush

    MHB 15.3 Express an integral for finding the area of region bounded by:

    ok so there are 3 peices to this Express and integral for finding the area of region bounded by: \begin{align*}\displaystyle y&=2\sqrt{x}\\ 3y&=x\\ y&=x-2 \end{align*}
  24. M

    Newtonian energy integral and suitable boundary conditions

    I have a (somewhat) strange energy equation which has the following form: KE = A + B W + C \exp(-D W), where A,B,D are known constant, C is an unknown constant to be determined and kinetic and potential energy are given by KE and W respectively with W\equiv W(r) i.e. is a function of...
  25. B

    A Intuition on integral term in D'Alembert's formula

    If $$\phi(t,x)$$ is a solution to the one dimensional wave equation and if the initial conditions $$\phi(0,x) , \phi_t(0,x)$$ are given, D'Alembert's Formula gives $$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . \tag{1}$$ which is...
  26. karush

    MHB Double Integral: Evaluating $II_{5a}$ in $R=[0,2] \times [-1,1]$

    $\textsf{a. Evaluate :}$ \begin{align*}\displaystyle R&=[0,2] \times [-1,1]\\ II_{5a}&=\iint\limits_{R}xy\sqrt{x^2+y^2}\, dA \end{align*} next step? $$\displaystyle\int_0^1 \int_{-1}^1 xy\sqrt{x^2+y^2}\, dxdy$$
  27. karush

    MHB Evaluating Double Integral $II_{5d}$

    $\textsf{d. Evaluate :}\\$ \begin{align*}\displaystyle II_{5d}&=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{1}{(x^2+1)(y^2+1)} \, dy dx \end{align*}
  28. B

    Improper Integral of a Monotonic Function

    Homework Statement Let ##f: (1, \infty) \to [0,\infty)## be a function such that the improper integral ##\int_{1}^{\infty} f(x)dx## converges. If ##f## is monotonically decreasing, then ##\lim_{x \to \infty} f(x)## exists. Homework EquationsThe Attempt at a Solution This problem doesn't come...
  29. karush

    MHB 15.1.25 Evaluate the following double integral over the region R

    $\tiny 15.1.25$ $\textsf{Evaluate the following double integral over the region R}\\$ $\textit{note: the R actually is supposed be under both Integrals don't know the LaTEX for it}$ \begin{align*}\displaystyle \int_R\int&=5(x^5 - y^5)^2 dA\\ R&=[(x,y): 0 \le x \le 1, \, -1 \le y \le -1]...
  30. S

    A Analytical evaluation of r^l integral

    Hello, I need to find the matrix elements of in the particular case where l = 1. This should have an analytical solution but I have no idea where to start with this demonstration. Any suggestions on where to start digging?Ty!
  31. lfdahl

    MHB Calculate the integral ∫(tanx+cotx)(tanx/(1+cotx))^2dx.

    Calculate the integral: \[I = \int_{0}^{\frac{\pi}{4}}\left(\tan x + \cot x \right)\left ( \frac{\tan x}{1 + \cot x} \right )^2dx.\] A solution without the use of an online integral calculator is preferred. :cool:
  32. H

    Integral constant for internal energy of an ionic liquid

    Integral constant for internal energy of ionic liquid I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this...
  33. H

    Integral constant for internal energy of ionic liquid

    Integral constant for internal energy of ionic liquid I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this...
  34. Isaac0427

    I Program to help calculate an integral

    Hi, I have this integral that I really want to calculate for a personal project (not for school), so I typed it into WolframAlpha and it said that the it took too long to compute and to get it computed I would have to pay money. Is there any free software that may be able to calculate this...
  35. P

    I Gauss's law of sphere using integral

    Hey I was just practicing Gauss's law outside a sphere of radius R with total charge q enclosed. So I know they easiest way to do this is: ∫E⋅da=Q/ε E*4π*r^2=q/ε E=q/(4*πε) in the r-hat direction But I am confusing about setting up the integral to get the same result I tried ∫ 0 to pi ∫0 to...
  36. S

    How can I find the limit of the integral?

    Homework Statement Hi! I need to find the limit when x-> +infinity of (integral from x to x^2 of (sqrt(t^3+1)dt))/x^5 Homework EquationsThe Attempt at a Solution The integral of (sqrt(t^3+1)dt) can only be estimated, so sqrt(t^3+1)=(t^(3/2))*sqrt(1+1/t^3) should I use the maclaurin series...
  37. DeathbyGreen

    A Pierels substitution integral approximation

    In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls...
  38. Math Amateur

    MHB The Integers as an Ordered Integral Domain .... Bloch Theorem 1.4.6 ....

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...
  39. Math Amateur

    I Integers as an Ordered Integral Domain .... Bloch Th. 1.4.6

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...
  40. redtree

    I Checking My Understanding: Lagrangian & Path Integral Formulation

    I note the following: \begin{equation} \begin{split} \langle \vec{x}| \hat{U}(t-t_0) | \vec{x}_0 \rangle&=\langle \vec{x}| e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} | \vec{x}_0 \rangle \\ &=e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} \delta(\vec{x}-\vec{x}_0)...
  41. A

    Classical Recommendation needed for integral calculus books

    I have learned an adequate amount of calculus including implicit, parametric differentiation as well as Upton second order differential equations in high school math course. It was really abstract and we were taught only how to solve mathematical problems. Now, I need to model those problems in...
  42. DavideGenoa

    I Differentiating a particular integral (retarded potential)

    Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...
  43. E

    Line Integral Notation wrt Scalar Value function

    I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...
  44. lfdahl

    MHB Definite integral challenge ∫cos2017xsin2017xdx

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

    MHB 242 .10.09.8 Express the integrand as a sum of partial fractions and evaluate integral

    $\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...
  46. O

    MHB Joint probabilty integral boundaries understanding.

    Hi, I have homework question that I'm trying to solve. But I can't understand the basis. Here is a picture of the question and what I have done: My question is, How do I set the boundaries for the integral? 1. If I want the whole squart. 2. To sum up areas. Assume I want to sum the areas, I...
  47. S

    I Path Integral in QM: Resolving Confusion on Causality

    Hello! I am reading a derivation of the path formulation of QM and I am a bit confused. They first find a formula for the propagation between 2 points for an infinitesimal time ##\epsilon##. Then, they take a time interval T (not infinitesimal) and define ##\epsilon=\frac{T}{n}##. Then they sum...
  48. Charles Link

    Insights An Integral Result from Parseval's Theorem - Comments

    Charles Link submitted a new PF Insights post An Integral Result from Parseval's Theorem Continue reading the Original PF Insights Post.
  49. Const@ntine

    B Integral of sin^2(ωt) for t

    Hi! I came across a proof in my physics textbook (amperage=wattage/area), and it contained this integration: ∫0 T sin2(ωt) dt The whole thing: 1/T∫0 T sin2(ωt) dt = 1/T(t/2 + sin2ωt/2ω)|T 0 = 1/2 I didn't remember how to integrate that, so I went back to check my notes, and look at it at...
  50. Jianphys17

    I Fredholm integral equation with separable kernel

    Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...
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