Integral calculus Definition and 155 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. Lotto

    How to calculate primitive functions on maximal intervals for periodic functions?

    Since ##f(x)## is continuous in ##\mathbb R##, it has a primitive function in ##\mathbb R## as well, so we have to define ##F(x)## also for points ## \frac{\pi}{2}+k\pi##. ##\lim_{x \to \left(\frac{\pi}{2}+k\pi \right)^-} F(x) =\frac{\pi}{2}+k\pi -\frac{\pi}{2\sqrt 2}+C_k ## ##\lim_{x \to...
  2. M

    IVP problem using Laplace transform and partial fractions

    For this problem, The solution is, However, I'm confused by the partial fraction decomposition of ##\frac{2}{s^4(s^2 + 1)}## I never done that sort of thing before. However, I think it would be done like this (Please correct me if I am wrong, the algebra is crazy here). ##\frac{2}{s^4(s^2 +...
  3. walterminator

    Thermodynamics: Cooling of a heated block of iron in contact with air

    Φ = 𝜑S (dQ/dt) = k*S*dT avec dT = (1500 + 273.15) - (40 + 273.15) = 1460 [k] avec dQ = - 0.24 [kg] * 440 [j/(kg * k)] * 1460 [k] = - 154176 [j] donc, -(154176)/dt = 40 [W/(k * m²)] * 0.0062 [m²] * 1460 [k] = 362.08 [W]
  4. WMDhamnekar

    I Computing the expectation of the minimum difference between the 0th i.i.d.r.v. and ith i.i.d.r.v.s where 1 ≤ i ≤ n

    Problem :Let ##X_0,X_1,\dots,X_n## be independent random variables, each distributed uniformly on [0,1].Find ## E\left[ \min_{1\leq i\leq n}\vert X_0 -X_i\vert \right] ##. Would any member of Physics Forum take efforts to explain with all details the following author's solution to this...
  5. A

    Moment of Inertia of a Disc: Is it paradoxical?

    In deriving the MOI of a ring about its center perpendicular to plane, our teacher said think of ring as made up of ##dm## units each at ##R## distance from the axis therefore the MOI becomes: $$I=R^2\int{dm}=MR^2$$ If disc is considered to be made up of rods of ##dx## thickness, in this manner...
  6. G

    A Meaning of Gauss' mean value theorem?

  7. akrill

    Finding the Centre of Mass of a Hemisphere

    Place hemisphere in xyz coordinates so that the centre of the corresponding sphere is at the origin. Then notice that the centre of mass must be at some point on the z axis ( because the 4 sphere segments when cutting along the the xz and xy planes are of equal volume) y2 + x2 = r2 We want two...
  8. I

    How can I calculate the cumulative mass of a disk using disk mass density?

    I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function: ρ(r)=ρ0*e^(-r/h) A double integral in polar coordinates should do, but im not sure about the solution I get.
  9. S

    A How to Solve This Differential Equation Analytically?

    Hello, can someone help me to solve the following differential equation analitically: $$\frac{2 y''}{y'} - \frac{y'}{y} = \frac{x'}{x}$$ where ##y = y(t)## and ##x = x(t)## br Santiago
  10. murshid_islam

    I Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##

    I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check? Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
  11. A

    I Calculating the surface area of a sphere using dA

    Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
  12. Tapias5000

    Integral ## \int _{ }^{ }\frac{1}{\sqrt{x^3+1}}dx ##

    I also don't understand how to get the descending factorials for this hypergeometric series, I also know that there is another way to write it with gamma functions, but in any case how am I supposed to do this? If I write it as a general term, wolfram will give me the result which leaves me...
  13. D

    B Are there mathematicians that dislike integral calculus?

    Solving integrals by hand is difficult and prone to errors, and the techniques such as integration by parts, partial fraction decomposition, and trig substitutions only work for a small subset of integrals and I do not see the point of avoiding technology like Wolfram Mathematica for...
  14. Safinaz

    Integration of an exponential function

    My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!
  15. TheGreatDeadOne

    I Using recurrence formula to solve Legendre polynomial integral

    I am trying to prove the following expression below: $$ \int _{0}^{1}p_{l}(x)dx=\frac{p_{l-1}(0)}{l+1} \quad \text{for }l \geq 1 $$ The first thing I did was use the following relation: $$lp_l(x)+p'_{l-1}-xp_l(x)=0$$ Substituting in integral I get: $$\frac{1}{l}\left[ \int_0^1 xp'_l(x)dx...
  16. U

    I Approximation of a function with another function

    Hi, I am wondering if it is possible to demonstrate that: tends to in the limit of both x and y going to infinity. In this case, it is needed to introduce a measure of the error of the approximation, as the integral of the difference between the two functions? Can this be viewed as a norm...
  17. 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} }##
  18. J

    Is this manipulation "evil" or simply not intuitive due to unfamiliarity?

    The solution is as follows : The substitution is what nags me , which is as follows : This substitution "trick" to me seems impossibly difficult to arrive at "logically" without pretty much reverse engineering the problem. So is this simply a lack of practice/ familiarisation showing ? I feel...
  19. CaptainX

    I Surface Integrals: Finding Centroid & Inertia of Circle

    How to find the centroid of circle whose surface-density varies as the nth power of the distance from a point O on the circumference. Also it's moments of inertia about the diameter through O. I'm getting x'=2a(n-2)/(n+2) And about diameter -4(a×a)M{something}
  20. A

    Troubles with an antiderivative

    please tell me if i am right or wrong
  21. S

    Comparing summations with integrals

    1. ##\sum_{n=1}^N \arctan{(n)} \geq N \arctan{(N)}-(1/2)\ln{(1+N^2)} \iff \sum_{n=1}^N \arctan{(n)} \geq N \int_0^N \frac{1}{1+x^2} dx - \int_0^N \frac{x}{1+x^2} dx## Where do I go from here? I've tried understanding this graphically, but to no avail. 2. Maybe this follows from finding an...
  22. S

    Checking for integrability on a half-open interval

    For a closed interval ##[a,b]## I have learned that ##U(f,P)-L(f,P)=\frac{(f(b)-f(a))\cdot(b-a)}{N}## where ##N## is the number of subintervals of ##[a,b]## (if ##f## is monotonically decreasing, change the numerator of the fraction to ##f(a)-f(b)##). However, if the interval is half-open, then...
  23. S

    Riemann sums for discontinuous functions

    The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...
  24. Drioton

    How do I find the area of the region bounded by following?

    Using integrals, consider the 7 requirements: Any my attempted solution that I have no idea where I am going: And the other one provides the graph:
  25. A

    What is the Correct Approach to Solve This Complex Integral?

    I split this to get \begin{equation} \int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx \end{equation} Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...
  26. JD_PM

    Trying to find an alternative for solving an integral

    Homework Statement Homework Equations The Attempt at a Solution [/B] I understand how the integral is solved using cartesian coordinates. However, I wanted to try to solve it using polar coordinates: $$\int_0^{\pi/2} cos \theta \sqrt{1+r^2 cos^2 \theta}d...
  27. Zack K

    I Why can an infinite area have a finite volume or SA?

    I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it. I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.
  28. JD_PM

    Changing the order of a triple integral

    Homework Statement $$\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) dxdzdy$$ Change the order of the integral to $$\iiint f(x,y,z) \, \mathrm{dydzdx}$$What I have done It is just about: From ##x=0## to ##x=\sqrt{4y+23}## From ##z=0## to ##z=4-y## From ##y=\frac{x^2-23}{4}## to...
  29. JD_PM

    What is the volume inside an ellipsoid between two intersecting planes?

    Homework Statement Find the volume between the planes ##y=0## and ##y=x## and inside the ellipsoid ##\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1##The Attempt at a Solution I understand we can approach this problem under the change of variables: $$x=au; y= bv; z=cw$$ Thus we get...
  30. S

    Area of a bounded region using integration

    In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
  31. JD_PM

    Evaluate an integral over a triangle

    Homework Statement Let D be the triangle with vetrices ##( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 )##. Evaluate the integral : $$\iint_D e^{\frac{y-x}{y+x}}$$ Homework EquationsThe Attempt at a Solution [/B] The answer to this problem is known (...
  32. Zack K

    Area between 2 polar equations

    Homework Statement Find the area of the region that lies inside the first curve and outside the second curve. ##r=6## ##r=6-6sin(\theta)## Homework Equations ##A=\frac {1} {2}r^2\theta## The Attempt at a Solution \[/B] If I'm correct, the area should just be ##\frac {1} {2}\int_{0}^{2\pi} 6^2...
  33. EEristavi

    Finding Length of a Curve: y2 = (x-1)3

    Homework Statement I have to find length of the curve: y2 = (x-1)3 from (1,0) to (2,1) Homework Equations s = ∫ √(1 + (f '(x) )2 ) dx where we have integral from a to b The Attempt at a Solution I'm bit confused: I'm thinking of writing function regarding x, f(x)...
  34. F

    Simplifying integral question....

    Homework Statement I have solve the integral for:to be: But I cannot figure out how to simplify the answer to the form shown above. Homework Equations The Attempt at a Solution Here is my progress so far: Any help would be appreciated
  35. H

    ##\int (\sin x + 2\cos x)^3\,dx##

    Homework Statement $$\int (sinx + 2cos x)^3dx$$ Homework EquationsThe Attempt at a Solution $$\int (sinx + 2cos x)^3dx$$ $$\int (sinx + 2cos x)((sinx + 2cos x)^2dx)$$ $$\int (sinx + 2cos x)(1 + 3cos^2x+2sin2x)dx$$ How to do this in simpler way?
  36. K

    How Do Zeroes of a Function Relate to Its Integral?

    Homework Statement You are given the function f(x)=3x^2-4x-8 a) Find the values of a. Explain the answers using the function. Homework EquationsThe Attempt at a Solution a^3-2*a^2-8*a=0 a=-2 v a=0 v a=4 I found the answers, but I don't know how to explain my answers by using the function...
  37. Mutatis

    How to integrate this function?

    Homework Statement I've got to integrate the following $$ \int dx =\int \frac {d\phi} {\phi \sqrt {1 - \phi²}}. $$ Homework Equations [/B] I already know the answer but not how to get it. The answer that I got from solution is ## x = \operatorname {arcsech}{\phi} ##. The Attempt at a...
  38. Peter Alexander

    Uniform convergence of a parameter-dependent integral

    Hello everyone! I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...
  39. JorgeM

    I Is this expression for Mass Flowing Out of a Bottle correct?

    Hello there, I am going to implement a numerical method to solve how much Mass of a fluid has gone out of an opened bottle. I am going to get h(t) with some sensors but I am not sure if this expression for Mass gone out "M" is correct. The area of the bottle at the up part. A1 The density of the...
  40. bornofflame

    [MultiVarCalc] Find the volume of the solid

    Homework Statement Find the volume of the solid between the cone ##z = \sqrt{x^2 + y^2}## and the paraboloid ##z = 12 - x^2 - y^2##. Homework Equations ##x^2 + y^2 = r^2## The Attempt at a Solution I drew a simple diagram to start off with to visualize the solid formed by the intersection of...
  41. T

    A simple case of translation invariance of Riemann integrals

    Homework Statement Show that \int_{A} 1 = \int_{T(A)} 1 given A is an arbitrary region in R^n (not necessarily a rectangle) and T is a translation in R^n. Homework Equations Normally we find Riemann integrals by creating a rectangle R that includes A and set the function to be zero when x...
  42. Zack K

    Finding the Value of an Integral with U-Substitution

    Homework Statement Suppose that: 0∫2f(x)dx = 2 1∫2f(x)dx = -1 and 2∫4 = 7, find 0∫1f(x+1)dx Homework Equations a∫bf(x) = F(b) - F(a) The Attempt at a Solution So in these types of integration, we are needed to use u-substitution, the problem is, using u-substitution requires you to have...
  43. 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
  44. H

    Calculating Arc Length and Area of Cylinder Side

    Homework Statement Let r be a positive constant. [/B] Consider the cylinder x2 + y2 ≤ r2, and let C be the part of the cylinder that satisfies 0 ≤ z ≤ y. (3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point (r cos θ, r sin θ, 0) (0 ≤ θ ≤ π). Let...
  45. 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. :(
  46. 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...
  47. 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 $f(2x)$.
  48. 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 $$0 \le t\le 1$$.
  49. M

    I Looking for additional material about limits and distributions

    I would like some help to find some additional info on generalized functions, generalized limits. My aim is to understand the strict definition of delta dirac δ(τ).If you could provide a concise tutorial focusing on δ(τ) not the entire theory...it would be of great help. I am not a math...
  50. 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...
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