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.

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  1. T

    I Derivative of A Def. Integral Equals Another Def. Integral?

    I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...
  2. N

    MHB Evaluation of definite integral (is it correct?)

    Is this question correct? We are given to evaluate: \int_0^2 \left(e^x-e^{-x}\right)^2\,dx 2\left(\frac{1}{2}\sinh(x)-x\right) 2\left(\frac{1}{2}\sinh(2\cdot2)-2\right)-2\left(\frac{1}{2}\sinh(2\cdot0)-0\right) \sinh(4)-4
  3. C

    MHB What is the correct solution for $\int_{0}^{1}x\sqrt{18-2x^2} \,dx$?

    Hello, I need some help finding the integral $\int_{0}^{1}x\sqrt{18-2x^2} \,dx$ Let $u= 18-2x^2$ $du=-4xdx$ $-1/4 \int_{16}^{18} \sqrt{u} \,du$= $36 sqrt(2) -128/3$ I am getting the wrong solution: $9\sqrt(2)-32/3$
  4. N

    MHB Finding area (application of definite integral)

    Hi, I am stuck on this question and was wondering if anyone could help me. The topic is integral equations. A block of land is bounded by two fences running North-South 5 km apart a fence line which is approximated by the function N=0.5E and a road which is approximated by the curve...
  5. C

    Solving for I1/I2 using substitution

    Homework Statement Hi all, This problem has been troubling me for a while now; even though I have tried my best ( and filled up a rough notebook in the process). Consider $$I_1=\int_{0}^{1} \frac{tan^{-1}x}{x} dx$$$, and $$I_2=\int_{0}^{\pi/2} \frac{x}{sinx}dx$$. We are supposed to find...
  6. O

    Derivative of Definite Integral Conundrum

    Homework Statement The normal approach using the fundamental theorem of calculus seems inapplicable. I define a function B(R) based on a definite integral with one of the limits being R. One factor in the definite integral has R in it and that function vanishes to 0 at x = R. Using the...
  7. lfdahl

    MHB Can You Solve This Definite Integral Challenge with Binomial Expansion?

    Derive an expression for the definite integral:\[I = \int_{0}^{\frac{\pi}{4}}sec^m(x)dx, \;\;\;\;m = 2,4,6,...\]
  8. Prasun-rick

    I How do you solve the definite integral ∫|x^3-x|dx with limits from -1 to 2?

    ∫|x^3-x|dx with limits from -1 to 2.Can anyone kindly show the step wise solution or suggest how to proceed .Thanks in advance
  9. C

    I What is this one's definite integral ?

    This is too far behind me ( time-wise ), and I've gotten different answers from virtually every different online integral calculator . What is the integral of : ( 2.12068539072 x 10^-11 ) / x^2 from 6,378,100 to 1,500,000,000 ? Please only help if you really know your maths well .
  10. karush

    MHB LCC 205 8.4.7 sine substitution (definite integral)

    $\tiny{LCC \ \ 205 \ \ \ 8.4.7 \ \ sine \ \ substitution }$ $\displaystyle I=\int_0^{5/2} \frac{1}{\sqrt{25-x^2}}\ dx =\frac{\pi}{6} $ $$\displaystyle x=5\sin\left({u}\right) \ \ \ dx=5\cos\left({u}\right) \ \ du $$ $$\displaystyle I=\int_0^{5/2} \frac{1}{\sqrt{25-25 \sin^2...
  11. Saracen Rue

    B Making a definite integral equal and indefinite integral?

    I have a calculator which allows me to sketch indefinite integrals - it assumes c = 0. However, when I try to use Desmos Online Graphing Calculator, it won't let me do this with it's integral function. It keeps trying to make me use definite integrals. I know that ∫(a,b,f(x)dx = F(a) - F(b), so...
  12. J

    MHB Efficient Techniques for Solving Complex Integrals | Help with Definite Integral

    I'm having a tough time with this integral: $$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$ where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck. What contour should I...
  13. J

    I Need help with this definite integral

    I'm having a tough time with this integral: $$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$ where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck. What contour should I...
  14. Sahil Kukreja

    Question on definite integral and inflection points

    Homework Statement Let P(x) be a polynomial of least degree whose graph has three points of inflection (-1,-1) ; (1,1) and a point with abscissa 0 at which the curve is inclined to the axis of abscissa at an angle of ## \frac {\pi}{3} , Then \int_0^1 P(x) \,dx = ? ## Homework...
  15. NatFex

    I Sum of Probability Density Function > 1?

    I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
  16. lep11

    Prove value of definite integral is greater than b

    Homework Statement Prove ##\int_{0}^{b} \frac{e^x}{1+x} dx>b## for every ##b>0##. The Attempt at a Solution Honestly, no idea. I suppose I cannot just calculate the definite integral and estimate. I think it needs to be proven in a more rigorious way.
  17. T

    I Derivative of a definite integral?

    consider x is between the interval [a,b] would it be correct to say that the derivative of a definite integral F(x) is f(x) because as dx approaches zero in (x + dx), the width of ALL "imaginary rectangles" would closely resemble a line segment which approximates f(x)? therefore change in area...
  18. JulienB

    Proof that the definite integral of 1/ln(x) doesn't exist

    Homework Statement Hi everybody! I'm having a hard time to find a way to cleanly prove that ∫(1/ln(x)) dx between 1 and 2 doesn't exist. At first I thought it was because it's not bounded (Riemann criterion I believe), but then I looked at another unbounded definite integral such as ∫lnx dx...
  19. Z

    Partial Derivative of a Definite Integral

    I'm trying to find the partial derivatives of: f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt and I am completely lost, any help would be appreciated, thanks.
  20. Steve Turchin

    Definite integral using diagonalizatio

    Homework Statement ##\int_{-\infty} ^{\infty} dx \int_{-\infty} ^{\infty} dy \ \ e^{-3x^2+2xy-3y^2} ## Homework Equations ## \int_{-\infty} ^{\infty} dx \int_{-\infty} ^{\infty} dy \ \ e^{-x^2-y^2} = \pi ## The Attempt at a Solution ##3x^2-2xy+3y^2 = (x,y) \left( \begin{array}{ccc} 3 & -1 \\...
  21. O

    Definite Integral by Definition

    Homework Statement Let A be the area of the region that lies under the graph of f(x) = 2x 2 + 5 between x = 0 and x = 4. Find an expression for A using n rectangles. Then evaluate this expression. Homework Equations Answer is 188/3 h= (4/n) The Attempt at a Solution [/B] The problem I am...
  22. N

    How to evaluate this definite integral?

    Hello, I don't know how to approach this problem, provided in the image below: I am assuming that in order to solve this problem, you have to transform the 2nd integral into the form of the first integral, but I am not sure if that's even the way to solve it, and even if it was, I don't know...
  23. I

    Areas And Distances (Intro. to Definite Integral)

    Hey everyone, Today in my Calculus 1 lecture we covered Areas and Distances, which serves as a prequel to the definite integral in my book. I am confused on some notation the book uses, and I cannot seem to find a clear explanation anywhere that I look. n ∑ f(xi) ΔX ≅ A i=1 First, let me...
  24. N

    What are the definitions and properties of Riemann sums?

    Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...
  25. J

    MHB Possible title: How do I solve this definite integral evaluation?

    Evaluation of \displaystyle \int_{0}^{\frac{\pi}{4}}\frac{\ln(\cot x)}{\left[(\sin x)^{2009}+(\cos x)^{2009}\right]^2}\cdot (\sin 2x)^{2008}dx What I have Tried:: Let \displaystyle \int_{0}^{\frac{\pi}{4}}\frac{\ln(\cot x)}{\left[(\sin x)^{2009}+(\cos x)^{2009}\right]^2}\cdot (\sin 2x)^{2008}dx...
  26. O

    Definite integral involving partial fractions

    Homework Statement Homework Equations trigonometric identities The Attempt at a Solution I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2 dθ = 2/(1+u2) du = 1/2 sec2(θ/2) dθ I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...
  27. B

    MHB Stumped by Definite Integral: $\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx$

    Was asked to solve this definite integral in a tech free test. Not sure how to go about it. $$\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx.$$ I know here is a relationship between inverse sin and the sqrt function but with just sin x?
  28. P

    Definite integral of an absolute value function

    Can we integrate: $$\int_a^b |x| dx$$ using an antiderivative of ##|x|##, namely ##\frac{1}{2} x |x|##, instead of splitting up the integration interval? I know this is not particularly useful for integrals such as: $$\int_{-5}^5 |t^3 - 8| dt$$ However, for absolute value functions with linear...
  29. Byeonggon Lee

    Definite integral of step function

    I need to prove whether this expression is true or false: ## \sum\limits_{k=1}^{n}\int_{k-1}^{k}[x]dx = \frac{n(n-1)}{2} ## I'm so confused because as I know, definite integral is possible only when the target function is continuous in closed interval. In this case, function ##[x]## should be...
  30. 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...
  31. S

    Integration by parts, just need a small hand

    Homework Statement I'm going to cut from the initial part of the problem, which I am confident is good to go, and cut straight to the antiderivatives. Homework Equations All antiderivatives are to be integrated on the interval from 0 to π/18 (I1) = -1/9 cos 9x - (I2) (-2/27 * cos3(9x)) + (I3)...
  32. Emmanuel_Euler

    Indefinite and definite integral of e^sin(x) dx

    Look to this indefinite integral →∫e^(sin(x))dx Antiderivative or integral could not be found.and impossible to solve. Look to this definite integral ∫e^(sin(x))dx (Upper bound is π and Lower bound is zero)=?? my question is : can we find any solution for this integral (definite integral) ??
  33. andyrk

    Definite Integral of Definite Integral

    h(x)= \int_0^x (\int_0^uf(t)dt). du, then why is h'(x) = \int_0^uf(t)dt? Shouldn't it be ## h(x) - h(0)## in the first equation? where ##h(x)## is the antiderivative of \int_0^uf(t)dt? But wait, isn't antiderivative of a function without limits on it? Like for \int_a^bf(x)dx we would say, let...
  34. A

    Find values for z for which the function f grows

    Member warned about not showing an attempt. 1. Homework Statement As the title says, I am supposed to find values for x for which the function given below grows. f(x)=(integral from -3 to x of t^4*e^(t^2)dt)+(integral from x^2 to 2 of t*e^tdt) Homework EquationsThe Attempt at a Solution I...
  35. Aceix

    Integrating a Definite Integral with an Undefined Function at One Endpoint

    Homework Statement integrate from 1 to 2 x(x^2-3)^(1/2) with respect to x. Homework EquationsThe Attempt at a Solution i attempted using numerical approximations but at x=1, the function is not defined so is there a way to combine improper integrals with this? Aceix.
  36. andyrk

    Definite integral of greatest integer function

    Why is π/2∫3π/2[cos-1t].dt = π/2∫3π/2-1.dt? [.] denotes the greatest integer function. Is it because [cos-1t] = -1 in t ∈ [π/2,3π/2)? But [cos-1t] = 0 for x = 3π/2. So we have one 0 along with all the -1's in (π/2,3π/2]. So how can we substitute [cos-1t] = -1 for every x in (π/2,3π/2] even...
  37. AdityaDev

    How to Simplify the Integral of (sinx-cosx)ln(sinx) from 0 to π/2?

    Homework Statement $$\int_0^{\pi/2}(sinx-cosx)ln(sinx)dx$$ Homework Equations ##int_0^af(x)dx=int_0^af(a-x)dx## The Attempt at a Solution Using above equation, you get (without integral sign): ##(sinx-cosx)ln(tanx)## but it did not make any difference. I got the answer by splitting the...
  38. J

    Is the definite integral a special case of functionals?

    So yesterday I learned about functionals, which my book claims are "machines that take a function and return a number", in contrast to functions, which take a number and return another number. I immediately thought of definite integration: it's an operation that takes a function, and returns a...
  39. K

    Definite Integral: Solving $y=\int^{10}_2 \frac{13.2}{x^{1.4}}$

    Homework Statement There is a problem in physics. i need to calculate the definite integral: $$y=\int^{10}_2 \frac{13.2}{x^{1.4}}$$ Homework Equations $$\int x^{-a}=\frac{1}{-a+1}x^{-a+1}$$ The Attempt at a Solution $$y=\int^{10}_2 \frac{13.2}{x^{1.4}}=13.2\int^{10}_2 x^{-1.4}=13.2...
  40. anemone

    MHB Can You Evaluate This Definite Integral with Trig Functions?

    Evaluate the definite integral $\displaystyle \int_{-\pi}^{\pi} \dfrac{\sin nx}{(1+2^x)\sin x}\,dx$, where $n$ is a natural number.
  41. H

    Curious definite integral : sine integral times exponential

    Hello PF, I just found a curious integral. I wondered if it comes from a bigger group of integral definitions: \int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a) Where Si(x) is the sine integral function \mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x I proved the...
  42. H

    Curious definite integral : sine integral times exponential

    Hello, and thanks for welcoming me in the forum of Physics Forums. I just found a curious integral that I solved by Taylor series. I wondered if it comes from a bigger group of integral definitions: ##\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)## Where Si(x) is the sine...
  43. K

    Definite integral doesn't solve with L'Hopital

    I want to solve: $$\int_0^\infty \frac{dx}{\left( x^2+r^2 \right)^{3/2}}=\left[ \frac{x}{r^2\sqrt{x^2+r^2}}\right]_0^\infty$$ I apply L'Hopital's to the denominator: $$\left(r^2\sqrt{x^2+r^2}\right)'=\frac{xr^2}{\sqrt{x^2+r^2}}$$ I apply again and agin L'Hopital to this but all the time almost...
  44. B

    MHB Definite integral on elliptic integral where modulus is function of variable

    How to prove: $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $ where \[ k(\theta)=\sqrt\frac{4Rh \tan \theta}{Z^2+(R+h \tan \theta)^2}\] and $ K[k(\theta)] $ is the complete elliptic integral of the first kind...
  45. 0

    Calculate a definite integral from another definite integral

    Homework Statement f is continuous in [-1, 1]. Calculate ##\int_0^1 f(2x - 1) dx##, given that ##\int_{-1}^1 f(u) du = 5## f is continuous in [0, 4]. Calculate ##\int_{-2}^2 xf(x^2) dx## The Attempt at a Solution I did one easier exercise where both integrals were in terms of x, a simple x -...
  46. C

    Definite integral and constants

    Homework Statement If acceleration=dV/dt, V=def.integal from t1 to t2 of a*dt,=at+C. At t=0, C=V0. Why do we have a constant C, as a definite integral was calculated. Won't C2 and C1 cancel out? Homework Equations Definite integral from x1 to x2 of dx is x2-x1 without any constant, right? The...
  47. V

    How do I accurately compute a definite integral with a branch point?

    I am trying to compute the value of $$ \int\limits_{-1}^0 {dx \over \sqrt{\left(x + 1\right)\left(c^2 - x^2\right)}}, $$ where ## c^2 > 1 ##. This integral exists because the integrand is positive and ##c^2 - x^2 > c^2 - 1 = a^2 > 0 ##, so $$ \int\limits_{-1}^0 {dx\over \sqrt {\left(x + 1\right)...
  48. H

    Definite Integral of 1/x from 0 to 1

    Hello, Just wondering about something, given that ln(1)=0, then the below should hold true? \int_0^1 1/x=0 But the entire graph from 0 to 1 of 1/x is positive, unbounded in fact to positive infinity. So is there an intuitive explanation for why the area under the graph of 1/x from 0 to 1...
  49. J

    MHB Calculate Integral $\int_{0}^{1}x^{2014}\cdot \left(1-x\right)^{2014}dx$

    Calculation of Integral $\displaystyle \int_{0}^{1}x^{2014}\cdot \left(1-x\right)^{2014}dx$
  50. KleZMeR

    Evaluate Definite Integral with Complex Analysis

    Homework Statement I_1 = \int_0^{2\pi} \frac{sin\theta}{3+2cos\theta} d\theta Homework Equations Using identities to change from cos, sin, to variables of z, I get: 2iz^2 + 6iz + 2i in my denominator The Attempt at a Solution Looking for a singularity, will I use a quadratic...
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