# What is Integrals: 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. ### Find volume of this object using integrals

I am given this expression which represents an object in 3D and the goal is to determine its volume using multiple integrals. I started by drawing what I think is the object as well as two "slices" of that object on different planes (z=2 and z=1) I have tried using cartesian, cylindrical and...

7. ### POTW Gaussian Integrals in Two Dimensions

Let ##a, b##, and ##c## be real numbers such that ##a## and ##c## are positive and ##ac > b^2##. Evaluate the double integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-ax^2 - 2bxy - cy^2}\, dx\, dy$$
8. ### Confused about polar integrals and setting up bounds

So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds. Any help would be greatly appreciated! Thanks!!

10. ### POTW Exponential-Type Integrals

Evaluate, with proof, the definite integral $$\int_{-\infty}^\infty \frac{e^{ax}}{1 + e^x}\, dx$$ where ##0 < a < 1##.
11. ### A Approximating integrals of Bessel functions

I edited this to remove some details/attempts that I no longer think are correct or helpful. But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more...
12. ### I Why are there 2s -1 independent integrals of motion?

I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and it can be considered an additive constant of time. Hence I tried searching it up online...
13. ### I Properties of Line Integrals question

I don't have any idea to answer these questions. I am working on it by searching the reference books where similar questions have been solved by authors. Meanwhile, any member of Physics Forums may help me in answering these questions.
14. ### Statistical mechanics and problem with integrals

So we have a system of N non interacting particles, on a d-dimensional space, the system is in contact with a bath of temperature T. The hamiltonian is $$H = \sum_{l = 1}^{N} (A_{l}|p_{l}|^{s}+B_{l}|q_{l}|^{s})$$. What is the avarage energy? Now, i have some problems with statistical...
15. ### Using Multiple integrals to compute expected value

I want to know how did author derive the red underlined term in the following Example?
16. ### I Do volume integrals involve bounding surfaces?

In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents) \begin{align} \vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3}...
17. ### I I need integrals Int[0,infty]t^(a-1) e^(-t) F(z, e^(-t))dt

I’m doing some brainstorming for a note I’m writing, I would appreciate it if anybody knows interesting integrals of the form $$\int_{t=0}^\infty t^{\alpha - 1} e^{-t} F(z, e^{-t})\, dt=G( z, \alpha )$$ where ##z## and ##\alpha## are complex parameters and the solution ##G(z, \alpha )## is...
18. ### Change of variables in multiple integrals

Summary: Find the volume V of the solid inside both ## x^2 + y^2 + z^2 =4## and ## x^2 +y^2 =1## My attempt to answer this question: given ## x^2 + y^2 +z^2 =4; x^2 + y^2 =1 \therefore z^2 =3 \Rightarrow z=\sqrt{3}## ## \displaystyle\iiint\limits_R 1dV =...
19. ### Can Wolfram Alpha do n-d integrals symbolically?

The Q is all in the title. I’ve been working on Selberg integrals, curious if Alpha can help me?
20. ### I Question about Feynman’s path integrals

In Feynman’s path integrals, there is: ∫dq″Π0(t″,t′;q″,q′)=1 What is the funny pi looking symbol?
21. ### Change in the order of integration in triple integrals

If we solve the L.H.S. of this equation, we get ## \frac{(b-a)^3}{6}## and if we solve R.H.S. of this equation, we get ##-\frac{2b^3-3ba^2 +a^3}{6}## So, how can we say, this equation is valid? By the way, how can we use the hint given by the author here?
22. ### MHB Multiple Integrals

Find the volume V of the solid S bounded by the three coordinate planes, bounded above by the plane x + y + z = 2, and bounded below by the z = x + y. How to answer this question using triple integrals? How to draw sketch of this problem here ?
23. ### I Looking for an expert on integrals

I’ve written an insight article on what I think is original material (at least I’ve not seen it in my reading nor google): A Novel Technique of Calculating Unit Hypercube Integrals I am looking first for someone that can follow my work, I’ve had some mathematicians look over it but none whose...
24. ### A Multiplying divergent integrals using Hardy fields approach

I wonder if the following makes sense. Suppose we want to multiply ##\int_0^\infty e^x dx\cdot\int_0^\infty e^x dx##. The partial sums of these improper integrals are ##\int_0^x e^x dx=e^x-1##. Now we multiply the germs at infinity of these partial sums: ##(e^x-1)(e^x-1)=-2 e^x+e^{2 x}+1##...

32. ### I How do a bunch of integrals make an n-simplex or an n-cube?

This question arises from Carroll's Appendix I on the parallel propagator where he shows that, in matrix notation, it is given...
33. ### Intro Physics Books for high school physics E&M [No integrals]

Can you recommend introductory physics book for high school that contains E & M ? It should not have any integrals.
34. ### MHB Doubt from multiple integrals

I'm having a problem solving this, My approach is solving $x_1$ as a variable and rest as constants first and then going on further. But it is getting too lengthy. Is there any short method?
35. ### Mathematica Loop integrals using Package-X 2.0 in Wolfram Mathematica

Hello everyone, I have trouble installing Package-X 2.0 to Wolfram Mathematica. It says that the package should be available at https://packagex.hepforge.org but this page does not open. I tried everything I could to install and load this package but it was all unsuccessful. It seems as the...
36. ### Checking convergence of Gaussian integrals

a) First off, I computed the integral \begin{align*} Z(\lambda) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}-\frac{\lambda}{4!}x^4\right) \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}\right) \exp\left(...
37. ### A Why physicists cannot renormalize all divergent integrals?

Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values? I mean, operations on divergent integrals are not a problem, and techniques...
38. ### MHB Check convergence of integrals

Hey! :giggle: I want to check if the following integrals converge or diverge. 1 . $\displaystyle{\int_0^{+\infty}t^2e^{-t^2}\, dt}$ 2. $\displaystyle{\int_e^{+\infty}\frac{1}{t^n\ln t}\, dt, \ n\in \{1,2\}}$ 3. $\displaystyle{\int_0^{+\infty}\frac{\sin t}{\sqrt{t}}\, dt}$ 4...
39. ### Path Integrals in Wolfram Mathematica

Hello everyone ! I am new to this site so I 'd better say hello to you all ! I am finishing my BR in physics and part of this ending is to deliver a thesis . Long story short I must compute path-integrals in SU(2) and SU(3) pure yang-mills fields . Problem is that i was never very good with...
40. ### A Elliptic integrals

Why this integral is called elliptic? I(k)=\int^{\frac{\pi}{2}}_0(1-k^2\sin^2 \varphi)^{-\frac{1}{2}}d \varphi
41. ### Showing that these two integrals are equal

Mentor note: The OP has been notified that more of an effort must be shown in future posts. These two are equal to each other, but I can't figure out how they can be that. I know that 2 can be taken out if its in the function, but where does the 2 come from here?
42. ### I Double integrals - do areas cancel?

If i do a double integral of 1.dxdy to find an area of an odd function eg. y=x from +a to -a i get zero because the area below the x-axis cancels with the area above the x-axis. If i do a double integral of a circle centred at the origin i get the area to be πr2 ; so why doesn't the area below...
43. ### MHB Can double integrals be interpreted as net change?

I understand that single integrals over a function can be interpreted as net change. Net change of the quantity between the bounds of the integration. But I am trying hard to understand if double integration can also be regarded as net change? That is, the net change in volume when the two input...
44. E

### B Orientation of double integrals

I learned that, because ##du \, dv = \frac{\partial(u,v)}{\partial(x,y)} dx \, dy##, if you set ##u=y## and ##v=x## then you get that ##dx \, dy = - dy \, dx##. And that the product of two differentials is a wedge product, which is antisymmetric. If coordinates are orthogonal, then ##dx \, dy =...
45. ### Curve integrals, del operator

My attempt is below. Could somebody please check if everything is correct? Thanks in advance!
46. ### I Confirming my knowledge on surface integrals

Hi, I want to make sure my understanding of calculating surface integrals of vector fields is accurate. It was never presented this way in a textbook, but I put this together from pieces of knowledge. To my understanding, surface integrals can be calculated in four different ways (depending on...