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. D

    Proof of Power Rule (Definite Integral) for Negative Integers

    This is not actually a homework problem. Rather, it is a problem from Courant and Robbins' What is Mathematics?, Chapter 8: "The Calculus", page 409-410. Homework Statement Prove that for any rational k =/= -1 the same limit formula, N → k+1, and therefore the result: ∫a to b xk dx =...
  2. K

    Definite Integral with u/du subsitition

    Homework Statement Evaluate the indefinite integral by the method shown in Example 5. (the example in the book is just using the u and du substitution) Homework Equations None. The Attempt at a Solution The question is right below the instruction and after that question is my...
  3. P

    Help with a definite integral property

    Hello, Saw this expression: ^{inf}_{-inf} \int e^{x^{2}} = 2 \times ^{inf}_{+0} \int e^{x^{2}} I am unable to understand this change of base. You can change from -inf to 0 and 0 to +inf but do not see how that equals to 2 * the integral from 0 to infinity. I hope someone will help me...
  4. P

    Question about definite integral and orthogonal functions

    Hello, I am just going through a book on calculus and understand that the definite integral can be interpreted as area under the curve. Now I am trying to figure out the orthogonality relationship between functions and this is normally defined (as far as I can tell from the internet resources)...
  5. N

    How to Calculate the Integral of x / sqrt(x²+1) from 0 to sqrt(3)?

    The problem is: ∫ x / sqrt(x²+1) Upper limit is sqrt 3 lower limit is 0 I choose u as x²+1 1/2du = dx.x 1/2∫ 1/u^1/2 = 1/2∫ u^(-1/2)du = 1/2 ∫ U^(1/2) / (1/2) = (1/2) * (2/1) * u^1/2 = 1 * u^(1/2)u = (sqrt3)² + 1 = 4 u = 0² + 1 = 1 than i substituted 1 * 4^1/2 - 1 * 1^(1/2) = 1 * 2 - 1...
  6. J

    The area of a definite integral

    Homework Statement The problem requires the computation of an indefinite integral but says to "interpret as areas" which basically changes the question to the following; What is the area under the curve of the function y = root( 6 - 2x2 + 4x) from x = -1 to x = 3 Homework Equations...
  7. S

    Definite integral of an even function

    Homework Statement Integrate the definite integral \int_{-2}^{2}{\frac{x^2}{4+x^6} dx Homework Equations The Attempt at a Solution (1) The integrand f is an even function, therefore: 2\int_{0}^{2}{\frac{x^2}{4+x^6} dx (2) I re-expressed the denominator as...
  8. C

    Establishing a definite integral is btw 12&24

    Homework Statement Suppose f(2)=7, f(4)=1, and f'(x)< 0 for all x. Assuming f^(-1) is differentiable everywhere, establish that 12 < Integral from 1 to 7, of f^(-1)(x)dx < 24 Homework Equations N/A The Attempt at a Solution I do not know where to begin... =/
  9. B

    Definite integral over random interval

    Hi, all, assuming a and b are random variables and their pdf f(a) and f(b) are known. then, how do I solve for the definite integral given as v=\int\limits_{a}^{b} g(x) dx, where g(x) is a function of x? or, how do I solve the pdf of v? Thanks a lot..
  10. 2

    Solving Definite Integral with Inverse Hyperbolic Identities

    Hey guys, I was doing some practice questions and this particular one has me stumped. The topic was on integration with inverse hyperbolic identities, and I was asked to give exact solutions for the following integral: \int\sqrt{4x^2 -1} dx between \frac{1}{2} and \frac{13}{10}...
  11. N

    Need help solving definite integral

    I need help on this one \int_0^{100} \sqrt{x}ln{x}dx I've tried integrating it by parts and ended up with the equation [(2/3)(x^3/2)(lnx) - (4/9)(x^3/2)] from 0 to 100 However, I can't plug it in because there's no value of ln(0). So, I'm at a loss right...
  12. beeresearch

    How do I find the definite integral of (a/x^2) * (1/sqrt(1-(2a/x))) with steps?

    Hi, Could someone show me how to find the definite integral from x to x = ∞ of the following expression?F*dx = [(a/x^2) * (1/sqrt(1-(2a/x)))] * dx Show steps please.. Thanks.. Steven
  13. C

    Is there a trick to solving this definite integral?

    Hi, For evaluating the entropy of a gaussian distribution, I need to evaluate this integral: \int^{\infty}_{-\infty}exp(x^2)x^2dx There is no analytical solution for the indefinite integral, but is there a trick for evaluating this particular definite one? Thanks a lot -Patrick
  14. S

    Integral: Find x=sin θ for 0-0.5

    Homework Statement Find using substitution x=\sin \theta \int _0^{0.5}\frac{x}{\sqrt{1-x^2}}dxHomework Equations integrationThe Attempt at a Solution \int _0^{0.5}\frac{x}{\sqrt{1-x^2}}dx =\int_0^{\frac{\pi}{6}}\sin \theta\;d\theta =1-\frac{1}{2}\sqrt{3} What I want to ask is about changing...
  15. Z

    Calculating the Definite Integral of Sin^3(x)/x^3: Step-by-Step Solution

    Homework Statement \int_{0}^{\infty} \frac{\sin^{3}(x)}{x^{3}}\,dx Homework Equations e^{ix}=\sin(x)+i\cos(x) \sin(x)=\frac{e^{ix}-e^{-ix}}{2i} The Attempt at a Solution \int_{0}^{\infty} \frac{\sin^{3}(x)}{x^{3}}\,dx=\int_{0}^{\infty} \frac{1}{x^{3}}...
  16. D

    Definite Integral and Summation Equivalence

    Can someone give me an explanation or possibly a proof that \int^{a}_{b}f(x)dx= \displaystyle\lim_{m\to\infty}\sum^{m}_{k=1}f(x^{*}_{k})\Delta x
  17. M

    Use the definition of the definite integral (with right hand rule) to evaluate

    Homework Statement Use the definition of the definite integral (with right hand rule) to evaluate the following integral from -3 to 2 \int(4x^2-9x+2)dx Homework Equations \int from a to b of f(x)dx = limit as n\rightarrow\infty of \sum f(xi)\Deltax. i = 1The Attempt at a Solution I found delta...
  18. T

    Evaluating Definite Integrals: \int_0^1 \sqrt{x} - x^3 dx Explained

    \int_0^1 \sqrt{x} - x^3 dx How do I evaluate this? what does \sqrt{x} -x^3 = ?
  19. L

    Finding definite integral (trigonometric)

    Homework Statement Hi guys ,, i have the following question (it's in the attachment) : Find m and M such that m <= x sin x <= M if 0 <=x <= pi. (Any reasonably good bounds will do, I am not asking for the best possible bounds.) Hence find bounds on the value of the integral[x sin(x),0 to...
  20. N

    Definite Integral Word Problem

    Homework Statement A rectangular pool is 10 meters wide and 25 meters long. The depth of water "x" meters from the shallow end of the pool is Part A. Write a left-hand Riemann Sum with 5 terms that approximates the volume of the water in the pool. Is your approximation an underestimate or an...
  21. D

    Complex analysis definite integral involving cosine

    Homework Statement integral 1/(a+cos(t))^2 from 0 to pi. Homework Equations cos(t)=1/2(e^it+e^-it) z=e^it dz/(ie^it)=dt The Attempt at a Solution int dt/(a+cos(t))^2 = int dz/iz(a2+az+az-1+z2/4 +1/2 +z-2/4) so with these types of problems I normally can factor this guy...
  22. B

    Why are two methods for evaluating an integral giving different results?

    I have a problem which involves finding the moment of inertia which involves evaluating: \int_0^1 x^2 \left(\frac{\pi}{2} - \sin^{-1} x\right)dx Now eventually, we end up having to evaluate the integral through integration by parts: \int_0^1 \frac{x^3}{\sqrt{1-x^2}} \ dx Now this is really an...
  23. C

    A Definite integral where solution. involves infinity - infinity

    Homework Statement Evaluate the integral. http://www.freeimagehosting.net/uploads/176e17ca2f.jpg Homework Equations http://www.freeimagehosting.net/uploads/3168397520.jpg The Attempt at a Solution http://www.freeimagehosting.net/uploads/3168397520.jpg I am stuck here as the book...
  24. C

    A Definite integral where solution. involves infinity - infinity

    Homework Statement Evaluate the integral. http://www.freeimagehosting.net/uploads/176e17ca2f.jpg Homework Equations The Attempt at a Solution
  25. G

    Solve Definite Integral: l & n Constants

    I'd like to analyze the integral in the attachment but I'm clueless on how to do it. I'd like to get the result and understand the method behind it. l and n are constants. Can anyone help?
  26. B

    Oopsie, issue with change of variables to evaluate definite integral

    I needed to evaluate the following integral (for constructing Chebyshev polynomials by an orthogonalisation process), but I just discovered that I'm having an issue with the change of variable technique:P The specific integral itself is unimportant as to the issue I'm having, but by means of an...
  27. W

    Using a power series to approximate a definite integral

    How do I go about finishing/calculating this? Homework Statement Use a power series to approximate \int\cos 4x\log x dx to six decimal places. (bounds are from pi to 2pi) Homework Equations The Attempt at a Solution So I broke down the equation first: \int\cos 4x\log x dx =...
  28. W

    Is there an optimal way to check complicated definite integrals?

    Hello! I am looking for a value of an integral \int^{\infty}_0 {r^{3-\epsilon} \over (r^2+N^2)^2}dr I have tried looking up a book by Gradshteyn and Ryzhik, however, its structure is quite complicated. Should I rewrite the integrand in some other non-obvious way to find it? Would you...
  29. E

    Definite integral of functions of two variables

    Homework Statement A pile of Earth standing on flat ground near an abandoned mine has height 13 meters. The ground is the xy-plane; the origin is directly below the top of the pile and the z-axis is upward. The cross-section at height z is given by x^2 + y^2 = 13 - z for 0 \leq z \leq 13 with...
  30. Mentallic

    Simple Definite Integral: Find the Area Bounded by a Curve

    Homework Statement Find the area bounded by the curve y=x^2-2, the y-axis, y=0 and y=1 I got an answer, but when I checked it using graphmatica, it was wrong (also the result is logically too large). I cannot see where I went wrong though, so could someone please help me spot the mistake...
  31. K

    Definite Integral = Area Under curve(s)

    I know the title can be a bit misleading, yet it's really close to what I want to ask. Today I came upon an integral at school. The really easy one :\int_{-1}^{1}x^3 dx. Of course, calculating the integral, you get 0. Yet, as far as I know, the way to get the area under the curve is by...
  32. J

    Definite Integral Length of vector r(t)

    Homework Statement Evaluate the integral length of r(t)=[tihat +t^2jhat]dt from 0 to 2 Homework Equations The Attempt at a Solution I think I should find the length of r(t) first which would be sqrt(t^2ihat+t^4jhat). However I'm not sure how I would integrate sqrt(t^2ihat+t^4jhat).
  33. N

    Mastering the Definite Integral: A Comprehensive Guide

  34. J

    Definite integral: exponential with squared exponent

    Hi, I'm trying to solve the following: f(x) = \int^\infty_{-\infty}ce^{yx-y^2/2} dy where c is a constant My only idea thus far was that since it is an even function, the expression can be simplified to: = 2c\int^\infty_0 e^{y(x-{1/2}y)} dy but I'm stuck here. Anyone know how to do...
  35. N

    Integrating Definite Integral: 3x^5 + 4x^4 + x^2

    there's a que which baffles me today,so i hope any1 of you could help me to clear my doubt,here is the que: integrate with limit 3 on top n 1 below (3x^5 + 4x^4 + x^2)/x^3 dx .. so here goes my sol please rectify me.. SOL: 3x^2 + 4x + x^-1 then integrate it x^3 + 2x^2..<<<<then...
  36. M

    How can the following integral be evaluated for x^2<1?

    The following integral came up in a paper I was reading recently. \int_0^{2\pi}\ln(1 + x^2 - 2x\cos\theta)d\theta The answer, for x^2<1, is zero. I'm not sure why. I tried writing it as a power series and showing that the integral for any given power of x vanishes, but it got too messy...
  37. Z

    Definite Integral Homework: Evaluate the Integral

    Homework Statement Evaluate the definite integral. Homework Equations \int_{1}^{2} ( 2e^{-4x} -\frac{1}{x^2} ) dx Answer given by the book: \frac{1}{2}(e^{-4}-e^{-8}-1) The Attempt at a Solution u = 4x; x = u/4; du = 4 dx; dx = du/4; \frac{1}{2}\int e^{-u} du - 4\int...
  38. S

    Definite Integral using Residue Thm

    Homework Statement Calculate the integral [ z^4/(1 + z^8) ] over negative infinity to positive infinity. Homework Equations Residue Theorem. Specifically for real-valued rational functions (on the real axis) where the denominator exceeds the degree of the numerator by at least two or...
  39. S

    Is There an Easier Way to Solve this Definite Integral?

    Homework Statement solve the integral: ∫_(-∞)^∞▒〖x^2 e^(-λ(x-a)^2 ) 〗 dx where λ and a are positive real constants The Attempt at a Solution I tried integration by parts with and without y-substitution but neither worked for me. Without substitution, I set up the integral to look...
  40. A

    Taking the derivative of a definite integral

    How do I do it? For example, if I have: \int_{0}^{\infty}sf(x) dx How do I take the derivative with respect to x? I was trying to derive the formula for an inverse laplace transform when I realized that I didn't know how to take the derivative of a definite integral.
  41. P

    Definite integral distance from volume

    Homework Statement If you have a water tower that is spherical with a radius 20m, how far from the bottom will the water level be if it is filled to 1/4 of it's capacity. [b]2. Homework Equations \int_-20^20pi(400-y^2)dy=33510.32164m^3=3,351,032.164Liters The Attempt at a Solution...
  42. L

    Solving Definite Integral: \int x^2 \sqrt{4-x^2} dx | Textbook Question Answered

    The question in my textbook was: \int_{0}^{2} x^2 \sqrt{4-x^2} dx I decided to just leave out the lower and upper limits for now, and just solve \int x^2 \sqrt{4-x^2} dx. (It's a bit long, but I assure you I did the work.) Upon making the substitution of x = 2 \sin \theta, I got it down...
  43. N

    Interval of the definite integral

    Homework Statement Let F(x) = \int^{x}_____________{0} x*e^(t^2) dt for x\in[0,1]. Find F''(x) for x\in(0,1). My only problem is the x, because the interval of the definite integral goes from 0 to x, and x is in the integral, even though the integral is with respect to dt. So I'd just like...
  44. P

    Definite integral involving base e and square root of variable as exponent

    Homework Statement Calculate \int_{0}^{4}\sqrt{x} e^{\sqrt{x}) The Attempt at a Solution At first I tried substitution, but this didn't bring me anywhere since the integral is not of the form \int f(g(x))g'(x) My attempt at integration by parts also leads to an endless loop...
  45. M

    Challenging improper & definite integral

    Evaluate \int_{0}^{\infty} x\ (e^{3x}-1)^{-1/3}\ dx and \int_0^1\frac{x-1}{\ln\, x} \; dx
  46. A

    How to test if a definite integral is finite or not?

    Suppose I have a complicated integral whose exact evaluation seems extremely difficult or may be even impossible, in such a case is there any way to tell if the integration result is finite or not? suppose the problem is \int_{-\infty}^{\infty} f(x;a,b) dx I think there might be some...
  47. C

    Calculating Work Done in Lifting a Bucket from a 60ft Deep Well

    A bucket that weighs 3 lb and a rope of negligible weight are used to draw water from a well that's 60 feet deep. Suppose the bucket starts with 37 lb of water and is pulled up by a rope at 2 ft/sec, while water leaks out of the bucket at a rate of 1/4 lb/sec. A. How long does it take for...
  48. N

    Deducing Curve Shape from Definite Integral Estimates

    I am asked to deduce the shape of a curve by knowing the following: The estimate of the definite integral for the area using the trapezium rule with 2 intervals of equal widths is above the real value. The estimate of the definite integral for the area using the trapezium rule with 4...
  49. J

    Evaluate the definite integral

    [SOLVED] Evaluate the definite integral [b]1. Homework Statement \int Sin(3t) dt; the boundaries are Pi/3 and zero. Homework Equations The Attempt at a Solution 3-cos(pi/3)^{}2-cos(0)= not the correct answer. Help!
  50. D

    Definite Integral & Limit: Proving a Relation

    Homework Statement Show that each limit is a definite integral. Lim (n \rightarrow \infty) of \sum \frac{n}{n^{2}+i^{2}} from i=1 to n Homework Equations Lim (n \rightarrow \infty) of \sum f(c)\Delta X from i=1 to n The Attempt at a Solution I can't really get this started, so...
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