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. Md. Abde Mannaf

    How can it's solve using Leibniz Rule or other rule

  2. thegreengineer

    Substitution method for finding an integral's interval changes

    Look, I was wondering if substituting the variable more than once is valid and hence the definite integral intervals change this way. Consider the following integral (I'm working for finding the volume of a solid of revolution): *\pi \int_{-3}^{5}3^{2}-(\sqrt{\frac{y+3}{2}}+1)^2dy Personally I...
  3. yangshi

    Integrating Compressible Flow Equations for V as a Function of x in MATLAB

    Homework Statement ( V1.4 A.4 C1 - (1/V) ) dV = dA / A C1 is a constant, V=f(x), A=.25*pi*(.0222 - x2) I'm trying to simplify the equation into a form with no integrals or derivatives, so I can put it into MATLAB to spit out an expression for V as a function of x. Or is it possible to put all...
  4. K

    Computing an integral -- any method

    Hi, I have been trying to find an integral ## \int_{-\infty}^{+\infty} \frac{ e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2} dx ##. I initially posted this question in the complex analysis forum since I felt it might be done using contour integration. However now I realize it might not be the best way...
  5. O

    Difficult Question in Calculus — limits and integrals

    Homework Statement (hebrew) : f(x) a continuous function. proof the following Homework Equations I guess rules of limits and integrals The Attempt at a Solution I've tried several approaches: taking ln() of both sides and using L'Hospitale Rule. Thought about using integral reduction...
  6. T

    Integration using u substitution

    Homework Statement Evaluate the integral of (x+1)5^(x+1)^2 Homework EquationsThe Attempt at a Solution I set my u=(x+1) making du=1dx. This makes it u*5^u^2. I integrated the first u to be ((x+1)^2/2) however I don't know what to do with the 5^u^2
  7. C

    MHB Integral calculus: stellar stereography

    Hi there, I have a question I'm stuck on. It is: Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analysed from a photograph. Suppose that in a spherical cluster of radius R the...
  8. D

    Fundamental theorem of calculus - question & proof verifying

    I understand that the fundamental theorem of calculus is essentially the statement that the derivative of the anti-derivative F evaluated at x\in (a,b) is equal to the value of the primitive function (integrand) f evaluated at x\in (a,b), i.e. F'(x)=f(x). However, can one imply directly from...
  9. D

    Question on a particular integral property

    I've been reading Wald's book on general relativity and in one of the questions at the end of chapter 2 he gives a hint which says to make use the following integral identity (for a smooth function in): F(x)=F(a)+\int_{0}^{1}F'(t(x-a)+a)dt Is this result true simply because...
  10. Peeter

    Integration by parts, changing vector to moment & divergence

    In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence: \begin{align}\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot \mathbf{J} ) d^3 x\end{align} He calls this change "integration by parts". If this...
  11. N

    What values of a make this set a type II region?

    Homework Statement Find all values of a such that this set is a type II region (i.e. the bounds of x can be represented as functions of y, while the bounds of y are constant valued) -1<y<0, Y<x<-y (union) 0<y<1, -y+a<x<y+a Homework EquationsThe Attempt at a Solution I arrived at a being any...
  12. N

    ChetIs the Integral Convergent?

    Homework Statement Find whether the integral is convergent or not, and evaluate if convergent. Homework Equations integral 1/sqrt(x^4+x^2+1) from 1 to infinity The Attempt at a Solution 1/sqrt(x^4+x^2+1)<1/sqrt(x^4) 1/sqrt(x^4)=1/x^2 which is convergent for 1 to infinity and is 1 therefore...
  13. thegreengineer

    Integral calculus: integral variable substitution confusion

    Recently I started seeing integral calculus and right now we are covering the topic of the antiderivative. At first sign it was not very difficult, until we started seeing integral variable substitution. The problem starts right here: Let's suppose that we have a function like this: \int...
  14. C

    MHB Integral Calculus Help - Differentiating e^(-bx^2)

    hi , i need some help in this integral e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinityI tried differentiating e^(-bx^2) twice and it came up weird , is there any other way to do it ?
  15. L

    MHB Integral Calculus Help Needed: \iiint_T\sqrt{x^2+y^2}z^4e^{z^4}dx\ dy\ dz

    Hi! I have some problems with the integral \iiint_T\sqrt{x^2+y^2}z^4e^{z^4}dx\ dy\ dz where T=\{(x,y,z)\in\mathbb{R}^3:\sqrt{x^2+y^2}\leq z\leq 1\} I have tried to change it to spherical and cylindrical coordinates but... nothing Can someone help me? Thanks
  16. J

    Applied Integral Calculus What did I do Wrong?

    This is a problem I've been working on... I did some things wrong, and syntactically incorrectly, and just messy. Could you help me out? Tell me how I can write this more syntactically correctly, and cleanly, please? Also, I realize that some might believe this would be more appropriately put...
  17. K

    Trig Integral Calculus: Mastering Problem Solving Techniques

    Am I doing good on this problem?
  18. micromass

    Calculus Differential and Integral Calculus by Courant

    Author: Richard Courant Title: Differential and Integral Calculus Amazon Link: https://www.amazon.com/dp/4871878384/?tag=pfamazon01-20 https://www.amazon.com/dp/487187835X/?tag=pfamazon01-20 Prerequisities: Table of Contents of Volume 1: Introductory Remarks Introduction The Continuum of...
  19. V

    The Difference Quotient and Integral Calculus

    I'm just a high school senior who noticed that the derivative has a general formula that we all know is, \frac{f(x+h)-f(x)}{h} but that there is no general formula (at least I haven't heard of it yet) for the integral of a function. I know I cannot simply just take the inverse of the difference...
  20. P

    MHB Maybe add this to the Integral Calculus tutorial....

    Reading the Integral Calculus tutorial, I felt like contributing. Do with it what you wish... Proposition: You can evaluate areas exactly using Integrals. Knowing that an integral is an antiderivative, and that derivatives are RATES, it seems odd that going in reverse would give geometric...
  21. Ackbach

    MHB Integral Calculus Tutorial

    1. Prerequisites In order to study integral calculus, you must have a firm grasp of differential calculus. You can go to http://mathhelpboards.com/calculus-10/differential-calculus-tutorial-1393.html for my differential calculus tutorial, although, of course, there are many books and other...
  22. M

    Integral Calculus Homework: Mean & RMS Voltage

    Homework Statement A generated AC voltage has a value given by V = 4cos2θ. You will need to use the identity cos^2⁡θ = 1/2(1+cos2θ) a) Find the mean value of the voltage over a full cycle (0 ≤ θ ≤ 2π) using integral calculus b) Find the r.m.s. value of the voltage over a full...
  23. W

    How can I use all the data to solve this integral calculus problem?

    Deleted None
  24. Y

    Integral calculus involving Change of Variables Theorem

    Homework Statement Evaluate \iiint_\textrm{V} |xyz|dxdydz where V = \{(x,y,z) \in ℝ^3:\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ≤ 1\}Homework Equations Change of Variables Theorem: \int_\textrm{ψ(u)} f(x)dx = \int_\textrm{K} f(\Psi(u))|detD\Psi(u)|du Examples: 1) For a ball of...
  25. Y

    Integral calculus involving Fubini

    Integral calculus involving Fubini's Theorem Homework Statement f(x,y) = x + y, if: x^2 ≤ y ≤ 2x^2 f(x,y) = 0, otherwise Evaluate \iint_\textrm{I}f where I = [0,1] x [0,1] Homework Equations For a Jordan domain K in ℝ^n, let h: K → ℝ and g: K → ℝ be continuous...
  26. D

    Integral calculus: volume of a solid of revolution

    Homework Statement Find the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 that is revolved about the line x=1. The Attempt at a Solution dV=∏R2t where : t=dy R=1-(y-y3) =1-y+y3 so.. dV=∏(1-y+y3)2dy dV=∏(1-2y+y2+2y3-2y4+y6)dy V=∏∫ from 0 to 1 of...
  27. D

    Integral calculus: plane areas in polar coordinates

    what is the area inside the graph of r=2sinθ and outside the graph of r=sinθ+cosθ? so i compute for the values of 'r',... but, i only got one intersection point which is (45°, 1.41). there must be two intersection points right? but I've only got one. what shall i do? i cannot compute for...
  28. D

    Integral calculus: plane areas in rectangular coordinates

    Homework Statement Find the area between y= 1/(x2+1) and the x-axis, from x=0 to x=1 The Attempt at a Solution so when x=0, y=1 and when x=1, y=1/2 next i plot the points, so the intersection of the given equation is (0,1) and (1,1/2) Yh= Y-higher= 1/(x2+1) Yl= Y-lower= 0 the...
  29. M

    Integral Calculus - Trigonometric Substitution

    Homework Statement 2 ∫ dx / (x+1)√[2x(x+2)] 1 Homework Equations Let x = tan θ if √(a^2 + x^2) Where a = constant The Attempt at a Solution 2 ∫ dx / (x+1)√(2x)√(x+2) 1 2 1/√2 ( ∫ dx / (√x)(x+1)[√(x+2)] 1 Now make all x...
  30. D

    Questions With Integral Calculus

    Homework Statement find the following integrals. Homework Equations 1. ∫2x^3 sin(x^2)dx 2.∫ x(2x-3)^1/3dx The Attempt at a Solution Using U-Sub for number 2 i ended with ((3(2x-3)^7/3)/28)+((9(2x-3)^4/3)/16)+C I apologize for the form but I'm new here and don't really know...
  31. 1

    Just diving into integral calculus

    And the first thing I'm tackling is understanding the area of a shape by inscribed/circumscribed polygons. For example, the shape produced by the line: y = x + 1, the y axis, and the vertical line x = 2. Now, my understanding is that the area is the limit as n approaches infinity of the...
  32. X

    Integral Calculus Antiderivative Question

    Homework Statement Find the antiderivative of: \int \frac{dx}{x^3 - 2x^2 + 4x - 18} Homework Equations This was asked in my Calculus II class right after we finished dealing with Solving for Integrals using Partial Fractioning. The Attempt at a Solution This is really more of an algebra...
  33. T

    Integral Calculus inequalities problem

    Homework Statement Hey, just wondering how I might go about doing this problem, as I really have very little idea... Prove the following inequality: \frac{1}{e}\leq\frac{1}{4\pi^{2}}\int_{R}e^{cos(x-y)}dxdy\leqe (hopefully this reads "one over e is less than or equal to one over four pi...
  34. P

    Solve Integral Calculus: Find a Value for a

    Hi, I have another question that I am having trouble with: Homework Statement Find the value of a such that: \int_{0 }^{a } cos^2 (x) dx The Attempt at a Solution\int_{0}^{a}\frac{1}{2} (1+cos (2x)) \frac{x}{2} + \frac{(sin(2x)}{4} \frac{a}{2} + \frac{sin(2a)}{4} - 0 - 0 = 0.740...
  35. P

    Find Values of a for Integral Calculus Homework

    Homework Statement Find the values of a > 0, such that \int_{a }^{a^2} \frac{1}{1+x^2} dx = 0.22 The Attempt at a Solution I integrated it and got: \int_{a }^{a^2} \frac{1}{1+x^2} dx = 0.22 arctan(a^2) - arctan(a) = 0.22 I am stuck after this, how would I solve this :-/ Thanks
  36. R

    How Do You Model an Apple Falling into a Black Hole Using Integral Calculus?

    Just for fun I'd like to find the equation r(t) which defines the position of an apple as a function of time falling from a distance Ro (radius of the earth) down to a basketball with the mass of the earth--probably a black basketball hole. All websites show the equations for a distance where...
  37. R

    Understanding the Relationship of Integral Calculus: A vs. (x-A)f(x) = 0

    Hello all, I wanted to know whether the following two relations are same. \int x\;f(x)\;{\rm d}x = {\rm A} and \int (x-{\rm A})\;f(x)\;{\rm d}x = 0 are same? 'A' is some positive number, distribution function f(x) is normalized to 1. (i am trying to understand some details of first moment...
  38. stripes

    Integral calculus question, using limits.

    Homework Statement Let A be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2pi/n, show that A = (\frac{1}{2})nr^{2}sin(2(\pi)/n) Homework Equations Area of a circle = pi x r^2 Area of...
  39. Z

    Integral Calculus vs Topology vs ODE

    I'm a Physics/Math major- and am setting up my degree plan I've posted a similar thread before but now I only have one math elective left (and a boatload of choices, all of which sound interesting) I've narrowed it down to either: Integral Calculus, Topology, or Theory of Ordinary...
  40. H

    Introduction to calculus and differental and integral calculus - courant

    What's difference between those Courant's books? They both seems to be first year calc books.
  41. W

    Indefinite Integral Calculus II

    Homework Statement \int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx Homework Equations The Attempt at a Solution \int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx Okay I let... u=tanx du=sec^2x Then I got \int \frac{ u }{ (u+4)(u+2) } du Then...
  42. J

    Integral Calculus - reduction formula/formulae

    Hi everyone - first a thanks to all who helped me through differential calculus and limits etc - think I'm getting the hang of it... Anyway as you can probably guess the next topic is integration. I'm kinda stuck on the 'concept' of reduction formula. I've done the usual integration...
  43. D

    Calculating the Area of a Tetrahedron with Double Integral Calculus

    Consider the tetrahedron which is bounded on three sides by the coordinate planes and on the fourth by the plane x+(y/2)+(z/3)=1 Now the question asks to find the area of the tetrahedron which is neither vertical nor horizontal using integral calculus (a double integral)? I think they mean...
  44. M

    Application of integral calculus: Work (spring)

    hi there! I'm having some troubles regarding this question: "if a force of 5 pounds produces a stretch of 1/10 of the natural length, L , of the spring, how much work is done in stretching the spring to double its natural length?" i tried answering this but I'm not sure if my answer is...
  45. Q

    Integral Calculus: Integrating (secx)^2 - Learn How!

    It has been a while since I have had to do an integration such as this, it is probably quite simple lol. But could someone show me how to integrate (secx)^2, that is secx all squared. I know that the answer is tanx but i was wondering if someone could show me the method. Thank you.
  46. T

    Preparation for learning integral calculus

    I have taught myself derivatives and I have taught myself the power rule and substitution but the book that I have bin using doesn’t go more in depth than that so what should I learn next?
  47. A

    Basic Integral calculus volume problem

    Hi everyone, this is my first time posting on these forums. If I am doing anything wrong, please let me know. I am having a lot of trouble with conceptualizing rotating lines around the X- and Y-axis. The problem I am trying to visualize right now is... Using integrals to represent...
  48. L

    Applications of Integral Calculus to Root Solving

    As a Grade 12 student that is often required to find the roots of quadratics for math, physics, and chemistry problems, I wondered whether there would be any methods for solving these problems excepting the quadratic formula. I was pondering the implications of calculus in algebra and, although...
  49. E

    Differential and Integral Calculus I&II by R.Courant free online

    Hi there, I don't know if this has been posted before but I found an online version of both volumes. If I'm not mistaken these links have the full content of this classic set. http://kr.cs.ait.ac.th/~radok/math/mat6/startdiall.htm http://kr.cs.ait.ac.th/~radok/math/mat9/startall.htm
  50. N

    Modeling motion with air resistance (integral calculus)

    Please, any help would be appreciated. Air resistance is a force that acts in the direction opposite to the motion and increases in magnitude as velocity increases, let us assume at least initially that air resistance r is proportional to the velocity: r = pv, where p is a negative constant...
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