What is Definite integral: Definition and 390 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.

View More On Wikipedia.org
Recent contents Top users
  1. D

    A conjecture about the rationallity of a definite integral

    Is it true that \int_0^1 f(x) dx \in \mathbb{Q} \Rightarrow \int_0^1 x f(x) dx \in \mathbb{Q} ? (Suppose f(x) integrable as needed) I thought of this conjecture yesterday and still couldn't prove it, I tried using integration by parts to relate it to the original, but didn't...
  2. P

    Trigonometric Integration, Definite integral.

    Homework Statement Use 2 or more substitutions to find the following integrals hint : begin with u=cosx Homework Equations Integral 0--->pi/2 (cosxsinx)/swrt(cos2x+49 dx The Attempt at a Solution I'm still a little fuzzy on using multiple substitutions. From what I've read on...
  3. J

    Calculating Mass of a Subset in SO(n) Using Definite Integrals

    I'm wrestling with this problem and I'd appreciate any pointers anyone can give me. (I've tried Google, but no luck.) I'm trying to figure out the total mass, with respect to the Haar measure I guess, that's accounted for by a definite subset of \mathbf{SO}(n). Specifically, working in...
  4. S

    Help me to find a Definite Integral

    1. Evaluate [PLAIN]http://www.goiit.com/equations/2011/5/25/795493e8-6a4d-42b0-8c18-2afa4e75b653.png I have tried hard to solve it but I am not getting the exact answer. The ans is given to be -a^2/(1+a^2)
  5. F

    How Do You Calculate the Area Under a Complex Function?

    i am struggling with this question for quite few days now and i am nowhere near to an appropriate solution. The previous ( related) question were regarding the the function g(x)= 5+9x-2x^2 /(x^2 -6x+23)^3/2. i have found the derivatives, the stationary point and also to classify them...
  6. G

    Help evaluating Definite Integral

    Homework Statement Hi guys, need help evaluating this integral. Integrate from -1 to 1 (t^i)(t^5-t^4) dt Where i just an arbitrary variable, (it's actually an index number of a matrix) Homework Equations The Attempt at a Solution So I thought I would begin by expanding...
  7. R

    Definite Integral: Evaluate ∫1^8 (3 - y^(1/3) / y^(2/3)) dy | Problem #24"

    Evaluate each Definite integral. ∫ lower limit 1, upper limit 8. 3 - y^(1/3) / y^(2/3) ; dy Here's a link if you have trouble understanding my problem. It's #24. http://pic20.picturetrail.com/VOL1370/5671323/23643016/396407299.jpg I did ∫(3*y^(-2/3) - y^(-1/3)) dy from 1 to 8 9*y^(1/3) -...
  8. R

    Find Area Using Definite Integral: [5,10]

    Use the definite integral to find the area between the x-axis and F(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. F(x)= 4x-32 ; [5,10] I did y = 0 => 4x - 32 = 0 => x = 8 => graph crosses x-axis at x = 8 Also it is +ve for x > 8...
  9. I

    DEFINITE integral of sinaxsinbx FROM 0 to infinity

    Hello, first time poster. I am putting this question in the PDE section because this was a question I came up while solving a PDE question. Also, I figured that since this is not a straight homework question, I can post in this category. Mods, feel free to move this post to wherever you...
  10. T

    Definite Integral of 1/x from 0 to 1 problem

    Given that 1/x is symetric across y=x, why can't we say \int^1_0 1/x - x dx= \int^\infty_1 1/x + x dx? Geometrically, it makes sense, but ln(0) is clearly undefined.
  11. S

    Solve Definite Integral: п/2∫dx/(2+sinx)2=0

    Homework Statement п/2 ∫dx/(2+sinx)2 0 Homework Equations The Attempt at a Solution п/2 ∫dx/(2 + sinx)2 = 0 п/2 ∫dx/(4 + 4sinx + (sinx)2) 0 substitude t=tg x/2 => x=2arctgx dx=2/(1+t2) dt sinx=2t/(1+t2) t=tgx/2 => 1 ∫(1/(4 + 4*(2t/(1+t2)) +(2t/(1+t2))2 )) * (2/(1+t2))dt...
  12. S

    Solve Definite Integral: π/2 to 0

    Homework Statement π/2 d x/ ∫ (x2 - 2x cosx + 1) 0 п ∫ sin x. ex^2dx -п Homework Equations The Attempt at a Solution
  13. Z

    Integrating a Definite Integral with a Dummy Variable

    If we have a definite integral given by f(x) = ∫sintdt with limits from 0 to x, can we integrate the function directly by substituting t as x because the variable involved in the integrand is a dummy variable?
  14. B

    Using a Power Series to approximate a definite integral

    [PLAIN]http://img600.imageshack.us/img600/1210/11096142.png Hey I was wondering if you guys could help me out with this question... I think I have the right power series: = \frac{1}{1-x} + \frac{x}{1-x} = (1+x+x^{2}+x^{3}+...)(x+x^{2}+x^{3}+x^{4}+...) = 1+2x+2x^{2}+2x^{3}+2x^{4}+... = 1 +...
  15. Z

    A Property of Definite Integral

    Can somebody give me an example of a definite integral satisfying this property? :
  16. G

    What Is the Metric Space of the Domain and Range in Complex Integration?

    I've been studying complex analysis. In the part of complex integration, there is no reference about the metric space which a continuous function takes as its domain and range. So, i felt confused. Before asking my question, i use the definition of continuity as follows \rightarrow...
  17. L

    Evaluate a definite integral with two variables?

    Homework Statement Evaluate. \int y2(1 - y3/a2)-2 dy Upper limit = 0 Lower limit = -a Homework Equations The Attempt at a Solution The a's make me really confused and I have no idea how to even start this question. What I started with was: Set u = 1 - y3/a2 du =...
  18. L

    Challenging Definite Integral with Square Roots and Logarithms

    Homework Statement Evaluate integral from -1 to 1 of: sqrt(ln(6-x))/(sqrt(ln(6-x))+sqrt(ln(6+x))) Homework Equations n/a The Attempt at a Solution I barely know how to approach this integral. I've been trying to figure this out for a long time now, and I feel like I haven't...
  19. M

    Integration by parts for a definite integral

    [PLAIN]http://img25.imageshack.us/img25/8933/lastscante.jpg I am new to integration by parts and am not sure what boundries to use when eveluating v on the bottom right.
  20. R

    Integration by Parts definite integral

    Homework Statement The definite integral of from 0 to 1 of ∫ (r3)dr/sqrt(4+r2)Homework Equations ∫udv = uv - ∫vdu ∫du/sqrt(a2 - u2) = arcsin(u/a) + C ∫du/(asqrt(a2 - u2)) = (1/a)arcsec(u/a) + C The Attempt at a Solution I made u = (4+r2)-1/2 because I thought it easier to get it's...
  21. Z

    Definite Integral Homework Help - Sign Error?

    Homework Statement I am getting the answer as -(pi) The answer given is (pi). There might be a sign error. I checked it many times but cannot find my fault. I need some help.
  22. L

    Curious definite integral related to sophomores dream

    The integral I am seeking to evaluate is int_{-\inf}^{+\inf) exp(-|w|/2) exp( i w [ln(|w|)/pi-x]) dw a definite integral which is a function of x... at x=0, it is nearly the same as the "sophomore dream" integral... I don't know why, but this function seems to have a singularity at...
  23. C

    Derivative of a definite integral

    Homework Statement The problem: Find the derivative of x2 multiplied by the integral of dt/(t-2t2) from x to -x2+2x i.e, the function is x^2 times integral of [1/(t-2t^2)]dt from a to b with a being x and b being -x^2+2x Homework Equations Derivative and integral definitionsThe Attempt at...
  24. C

    Calc: Find a definite integral for the mass

    Homework Statement A cylinder mug with a 4 centimeter radius and 8 centimeter height is filled with tea. When sugar is added to the tea, the sugar usually settles to the bottom of the cylinder mug. The density of sugar in the tea at a height h centimeter from the bottom of the cylinder mug is...
  25. D

    Problem: Definite integral by subtitution

    Hi everyone, to day I have a problem in dealing with the following question: and I hope that you help me here is may question and my try: [PLAIN]http://img268.imageshack.us/img268/9291/scan0004ov.jpg wish you all the best
  26. T

    How do I fix my definite integral issue?

    Homework Statement Integrate from 0 to sqrt(pi/2). I can't figure out how to make the LaTeX work correctly for it, so it just looks like an indefinite integral. \int 7\theta^3cos(\theta^2) d\theta Homework Equations \int udv = uv - \int vdu The Attempt at a Solution First...
  27. G

    Differentiating a definite integral

    I was reading on a textbook and came across a step and I had no idea how the author got the next step from it. This was basically differentiation of a definite integral. So here it is. \frac{\partial}{\partial t}{\int_0^t}{\int_0^1}u_t^2(x,s)dx ds So I figured that u_t=\frac{\partial...
  28. K

    Integration: Evaluate the Definite Integral?

    Homework Statement Q) Evaluate The Definite Integral: ∫ (x^3) / (1 + x^4) dx Upper Limit: 1 Lower Limit: 0 Homework Equations The Attempt at a Solution I think I'm on the right track; u = 1 + x^4 du/dx = 4x^3 du = 4x^3 dx 1/4 du = x^3 dx When x = 0 ; u =...
  29. B

    Finding Function with Definite Integral & Arc Length

    Sorry, latex is being weird. I'm currently trying to come up with a way to find an equation that satisfies: s=\int_a^b \sqrt{(f'[x])^2+1} \, dx Which is arc length and G=\int_a^b f[x] \, dx which is area under the curve where A and s are known values, and f[a]=A, f[b]=B...
  30. Z

    Solving Definite Integral with a, b Parameters

    This is not really a homework. Homework Statement I am trying to solve one definite integral Homework Equations \int_0^{2 \pi} \frac{\sin^2{t}}{\sqrt{a\cos{t} + b}} dt where a, b are some positive numbers. The Attempt at a Solution I tried integrate by parts, also...
  31. A

    Can't remember how to do this simple definite integral

    If my Force = (-gamma)(dx/dt) velocity v = dx/dt Mass m (m)(dv/dt) = F = (-gamma)(dx/dt) Now I want to integrate it over kT<= t <= (k+1)T Rearranging gives me (dv/v) = (-gamma/m)dt The right side integrates to (-gamma/m)(T), but how do I integrate the left side over kT<= t <=...
  32. M

    Definite Integral: Exponential

    Homework Statement \int^{\infty}_{0} e^{-y} dy = 1 Homework Equations The Attempt at a Solution Why is this equality true? I understand that the integral and derivative of e^y is always e^y, but I can't make out why this definite integral is equal to 1. I graphed it using a...
  33. B

    Definite Integral (Square Root)

    Homework Statement [PLAIN]http://www4a.wolframalpha.com/Calculate/MSP/MSP184619c56eg84971107d00001e292h4340heh8g4?MSPStoreType=image/gif&s=35&w=135&h=53 It is a definite integral between x=0 and x=20. I forget how to integrate with a square root. Do you just change it to a power of a...
  34. Char. Limit

    Is this Definite Integral with Imaginary Bounds Real?

    So, I was playing on Wolfram Alpha, and I managed to come up with this: http://www.wolframalpha.com/input/?i=integral_%28+infinity+*+sqrt%28-1%29+%29%5Epi+e%5E%28ix%29+dx&x=0&y=0 In Tex, I believe this is... \int_{i\infty}^{\pi}e^{i x} dx = i However, I have more than one problem...
  35. N

    Definite integral of rational function

    Homework Statement The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2 Homework Equations The Attempt at a Solution First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review. I can think of a few things. I...
  36. C

    Evaluate this definite integral 20x + 3x^2 - (2/3)x^3 from [-2, 5] ?

    Homework Statement I get the answer 19/3 but the real answer is 343/3. Can you show work to how it is that? I did this many times and still came up with 19/3 instead I got up to evaluating the integral part of finding the area between two curves but can never get the correct answer. I...
  37. L

    Evaluate the definite integral for the area of the surface.

    Homework Statement Evaluate the definite integral for the area of the surface generated by revolving the curve about the x-axis. y=(x3/6) + (1/2x), [1,2] Homework Equations 2π∫[r(x)](1+[f'(x)2]) The Attempt at a Solution First I found the derivative. f'(x)= (x2/2) +...
  38. mnb96

    Conditions for existance of definite integral

    Hello, what are the conditions that a continuous even function f must satisfy in order to have \int_{-\infty}^{+\infty}f(x)dx < \infty ?
  39. stripes

    What is the Problem with the Definite Integral in this Equation?

    Homework Statement What is wrong with the equation: \int^{\pi}_{0} sec^{2} x dx = tan x\right|^{\pi}_{0} = 0 Homework Equations None The Attempt at a Solution I don't know where to begin. I am inclined to give a mathematical (as opposed to a paragraph-like) explanation as to...
  40. C

    What is the definite integral of 1/(36+x^2) with bounds [0, 6]

    Homework Statement What is the definite integral of 1/(36+x^2) with bounds [0, 6]? I've only been taught U substitution to handle problems like these. I let u = 36+x^2 and du=2xdx but I am stuck and don't know what to do. Te answer is pi/24 but I don't know how to obtain it. Is there a...
  41. A

    Extra Credit Definite Integral - No Solution In Book- Did I do This right?

    Homework Statement \int^{1}_{-1} 2x\sqrt{(1-x^{2})^{3} +1} dx The Attempt at a Solution This is one of those questions in a math book that is almost never assigned and can be seen as "going above and beyond" what the chapter teaches you. I kind of improvised on the solution by substituting...
  42. G

    Trouble with definite integral

    Homework Statement \int _{0}^{2\,\pi}\!\int _{0}^{1}\! 0.5\,{r}^{ 3.5}\sin \left( \theta \right) {\cos}^{2}+ 0.5\,{r}^{ 2.5}{\sin}^{2}+1/5\,{r}^{5}{\cos}^{2}{ dr}\,{d\theta} Homework Equations The Attempt at a Solution So I first integrate the r, which was relatively straight...
  43. D

    Solving a definite integral without integrating

    Orthonormal basis \big(\frac{1}{\sqrt{2}}, cos(2x)\big) \pi \int_{-\pi}^{\pi} sin^4(x)dx=\frac{3\pi}{4} Solve the integral without integrating. sin^4(x)=\big[\frac{1-cos(2x)}{2}\big]^2=\frac{1}{4} - \frac{2cos(2x)}{4} + \frac{cos^2(2x)}{4} I know I could rewrite cos^2(2x) but then I...
  44. Immersion

    Definite Integral: Limit of a Summation

    Homework Statement Hi guys, i have a exercise of the limit of a summation that is the formal definition of definite integral and i need resolve and explain, but i can't resolve for the rational exponent, for this, need help, thanks in advance. Homework Equations \lim_{n \rightarrow...
  45. N

    Evaluation of definite integral

    Homework Statement Need to evaluate int(-inf,inf)(x/(x^3+1)). Homework Equations The Attempt at a Solution I don't believe finding the residues will be a problem. However, the integral is over the whole real line, and there is a singularity at -1 on the real line, so I'm...
  46. B

    What is the Value of W in the Definite Integral Definition?

    Homework Statement The letter W is defined as . The value of W, correct to the nearest hundredth, is? Homework Equations The Attempt at a Solution Now, I had this question on a test and got it wrong. I get 0.93, which I think is correct. Apparently the answer is 0.24. I call BS...
  47. C

    Can the Definite Integral of sinc(x) Be Solved Using Fourier Techniques?

    I'm studying a course in Fourier. In a multi-choice question, one of the answers asks for the value of the definite integral of sin(ax)/x over [-pi,pi]. I am wondering if there is a way to calculate this integral (I guess using Fourier techniques) or not. It is possible that it can't be solved...
  48. Z

    Definite integral example: don't understand

    Homework Statement This is am example from my textbook \int_{1}^{3} x^2 dx = 26/3 It then goes to define the arbitrary partition in [1,3] and maximum / minimum for each subinterval blah blah.. then it says For each index i, 1 <= i <= n, 3x_{i-1}^2 \leq x_{i-1}^2 + x_{i-1}x_i + x_i^2...
  49. F

    Why Is the Definite Integral Zero for Symmetric Functions?

    Homework Statement The Attempt at a Solution I have no idea how to start this integral. Can anybody give me a hint to start it off? Thank you.
  50. 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 =...
Back
Top