What is Definite integrals: Definition and 78 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. S

    Pendulum Question (Matlab Coursework)

    Apologies for my lack of knowledge on the equations front, I have burnt by brain out on this and haven't the capacity to learn LaTex right now! So here's a screengrab of it: So, This is a Matlab coursework and I am struggling to work out how best to approach solving it. What I have so far is...
  2. D

    Solving a Problem: Have I Made a Mistake or Are the Solutions Wrong?

    Hi everyone To solve the below problem, I assumed the affected area was 2x2 minus the definite integral of the given function between 2 and 4. I then equated the answer for that with the given function to solve for a, b and c. I don't know why the solutions give b as 2ln5. Have I made a...
  3. benorin

    I I need integrals Int[0,infty]t^(a-1) e^(-t) F(z, e^(-t))dt

    I’m doing some brainstorming for a note I’m writing, I would appreciate it if anybody knows interesting integrals of the form $$\int_{t=0}^\infty t^{\alpha - 1} e^{-t} F(z, e^{-t})\, dt=G( z, \alpha )$$ where ##z## and ##\alpha## are complex parameters and the solution ##G(z, \alpha )## is...
  4. Saracen Rue

    B Why is the definite integral of 1/x from -1 to 1 undefined?

    I've always been taught that the indefinite integral of ##\frac{1}{x}## is ##\ln(|x|)##. Extending this to definite integrals, particularly over limits involving negative values, should work just like any other integral: $$\int_{-1}^{1} \frac {1} {x} dx = \ln(|-1|) - \ln(|1|) = \ln(1) - \ln(1)...
  5. J

    Definite integrals and Functionals

    Taken from Emmy Noether's wonderful theorem by Dwight. E Neuenschwander. Page 28 1. Homework Statement Under what circumstances are these definite integrals functionals; a) Mechanical work as a particle moves from position a to position b, while acted upon by a force F...
  6. H

    B Definite integrals with +ve and -ve values

    I understand that if you have a function in which you want to determine the full (i.e. account for positive and negative values) integral you need to break down your limits into separate intervals accordingly. Is there any way in which you can avoid this or is it mathematically impossible? If...
  7. doktorwho

    How to Prove the Integral Property for Definite Integrals

    Homework Statement Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving:: Homework Equations 3. The Attempt at a Solution [/B] As no...

    I Question about Complex limits of definite integrals

    Hi, I see a formula of gamma function and i have a question. (1) $$\Gamma (s) = \int_{0}^{\infty } e^{-x}\, x^{s-1} dx$$ (2) $$ x=a\, n^{p} \rightarrow dx=ap\, n^{p-1}dn$$ (3) $$\frac{\Gamma (s)}{pa^{s}} = \int_{0}^{\infty } e^{-an^{p}}\, n^{ps-1} dn$$ i understand the formula but...
  9. Cjosh

    Calculating the definite integral using FTC pt 2

    Homework Statement Sorry that I am not up on latex yet, but will describe the problem the best I can. On the interval of a=1 to b= 4 for X. ∫√5/√x. Homework EquationsThe Attempt at a Solution My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...
  10. Hamal_Arietis

    Calculating Definite Integrals of $$f_n(x)$$

    Homework Statement Given the function in x $$f_n(x)=sin^nx (n=1,2,3,...)$$ For this ##f_n(x)##, consider the definite intergral $$I_n=\int_{0}^{\pi/2}f_n(x)sin2xdx$$ a) Find ##I_n## b) Hence the obtain $$lim_{n→∞}(I_{n-1}+I_n+I_{n+1}+...+I_{2n-2})=\int_0^W\frac{X}{Y+x}dx$$ Find X,Y,Z. Homework...
  11. N

    Finding work done using definite integrals

    On applying definite integral to find work done, we integrate F.dx and apply lower and upper limits. Should we apply the dot product, before integration , that is -1 for θ = 180, 1 for θ = 0. Or will the limits applied and their values suffice in deciding the sign of the final value. I have...
  12. T

    B Trouble converting definite integrals to Riemann's and back

    can i request anyone to please show me the step by step with specific explanations? thank you! i saw this on stackexchange, and the steps shown are really fuzzy to me :(
  13. L

    Riemann's proof of the existence of definite integrals

    Hello, Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...
  14. 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...
  15. andyrk

    Mean Value Theorem for Definite Integrals

    In the MVT for Integrals: ##f(c)(b-a)=\int_a^bf(x)dx##, why does ##f(x)## have to be continuous in ##[a,b]##.
  16. P

    Definite Integrals: Prerequisites for Property

    I have a few questions about the following property of definite integrals: $$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$ What exactly are the prerequisites for this property? Should ##c## be a member of ##[a,b]##? Should the function ##f## be defined at ##c##?
  17. andyrk

    Newton Leibnitz Formula for Evaluating Definite Integrals

    Lately, I have been trying really hard to understand the Newton Leibnitz Formula for evaluating Definite Integrals. It states that- If f(x) is continuous in [a,b] then \int_a^b f(x) dx = F(b) - F(a). But one thing that just doesn't make sense to me is that why should f(x) be continuous in...
  18. titasB

    Integrating with Changing Intervals: Finding the Area Between Two Curves

    Homework Statement Find ∫ f(x) dx between [4,8] if, ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4 Homework Equations [/B] ∫ f(x) dx between [4,8] , ∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4 The Attempt at a Solution We are given ∫ f(2x) dx between...
  19. J

    MHB Evaluating Definite Integrals with Floor Function

    Evaluation of \displaystyle \int_{0}^{\pi}\lfloor \cot x \rfloor dx and \displaystyle \int_{0}^{\pi}\lfloor \cos x \rfloor dx\;, where \lfloor x \rfloor denote Floor function of x
  20. H

    Definite Integrals Using Contour Integration

    Problem Show: \int_0^\infty \frac{cos(mx)}{4x^4+5x^2+1} dx= \frac{\pi}{6}(2e^{(-m/2)}-e^{-m}) for m>0 The attempt at a solution The general idea seems to be to replace cos(mx) with ##e^{imz}## and then use contour integration and residue theory to solve the integral. Let ##f(z) =...
  21. L

    Definite integrals with trig issues

    from 0 to π/2 ∫sin5θ cos5θ dθ I have been trying to solve the above for quite some time now yet can't see what I am doing wrong. I break it down using double angle formulas into: ∫ 1/25 sin5(2θ) dθ 1/32 ∫sin4(2θ) * sin(2θ) dθ 1/32 ∫(1-cos2(2θ))2 * sin(2θ) dθ With this I can make u = cos(2θ)...
  22. L

    Evaluating definite integrals for the area of the regoin

    Homework Statement Evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. f(x) = \frac{1}{x^2+1} at the point (1,1/2) Homework Equations The Attempt at a Solution So far I...
  23. E

    Equality of definite integrals, relation between integrands

    Suppose we are given two functions: f:\mathbb R \times \mathbb C \rightarrow\mathbb C g:\mathbb R \times \mathbb C \rightarrow\mathbb C and the equation relating the Stieltjes Integrals \int_a^\infty f(x,z)d\sigma(x)=\int_a^\infty g(x,z)d\rho(x) where a is some real number, the...
  24. Saitama

    MHB Comparing fractions with definite integrals

    Hello! I found the following problem on AOPS: Which is larger, $$\Large \frac{\int_{0}^{\frac{\pi}{2}}x^{2014}\sin^{2014}x\ dx}{\int_{0}^{\frac{\pi}{2}}x^{2013}\sin^{2013}x\ dx}\ \text{or}\ \frac{\int_{0}^{\frac{\pi}{2}}x^{2011}\sin^{2011}x\ dx}{\int_{0}^{\frac{\pi}{2}}x^{2012}\sin^{2012}x\...
  25. J

    Calculus Definite Integrals: Volumes by Washer Method

    Homework Statement Using Washer Method: Revolve region R bounded by y=x^2 and y=x^.5 about y=-3 Homework Equations V= integral of A(x) from a to b with respect to a variable "x" A(x)=pi*radius^2 The Attempt at a Solution pi(integral of (x^.5-3)^2 -(x^2)^2-3) from 0 to 1 with...
  26. S

    MHB What is the solution to the definite integral $\int^1_0 x^2 e^x \, dx$?

    Evaluate the following integrals. a) $\int^1_0 x e^x dx$ So integrating by parts we get $u = x $ $vu = e^x dx$ $du = dx$ $ v = e^x$ $uv - \int vdu = x e^x - \int^1_0 e^x dx$ xe^x - e^x |^1_0 = 1 b) \int^1_0 x^2 e^x \, dx Integrating by parts we get u = x^2 dv = e^x dx du = 2xdx...
  27. C

    Antiderivative Definite Integrals

    Homework Statement So I did an entire antiderivative, and ended with this part: sec(x)tan(x) + ln|sec(x) + tan(x)| + C I have to do this with the lower bound of -pi/3 and 0. When I do it, I should be getting 2√3 + ln(2+√3) But, I'm getting (0+0)-(2*-√3 + ln(2-√3)) Which would...
  28. MarkFL

    MHB Angelina Lopez's questions at Yahoo Answers regarding definite integrals

    Here are the questions: I have posted a link there to this thread so the OP can see my work.
  29. S

    MHB Evaluating definite integrals via substitution.

    Can someone make sure I'm on the right track with this problem? I'm a little confused because I thought that when you make a substitution you update the limits and get better numbers to work with when you plug them in the function in the end...Yet, it seems like I almost got worse numbers to...
  30. T

    Application of Residue Theorem to Definite Integrals (Logarithm)

    I've been studying for a test and have been powering through the recommended problems and have stumbled upon a problem I just can't seem to figure out. $$\int_{0}^{\infty} \frac{logx}{1+x^{2}} dx$$ (Complex Variables, 2nd edition by Stephen D. Fisher; Exercise 17, Section 2.6; pg. 167)...
  31. Y

    Calculate definite integrals with given interval.

    I just want to verify is this the way to calculate the result of a definite integral with the given interval. Say the result of the integral over [0,##\frac{\pi}{2}##] is \sin(\theta)\cos(\theta)d\theta|_0^{\frac{\pi}{2}} It should be...
  32. polygamma

    MHB Integrals: Show $\alpha$ Not Int Multiple of $\pi$ and $s, \lambda >0$

    1) Show that for $\alpha$ not an integer multiple of $\pi$, $\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \Big( \frac{\cos x + \sin x}{\cos x - \sin x} \Big)^{\cos \alpha} \ dx = \frac{\pi}{2 \sin \left(\pi \cos^{2} \frac{\alpha}{2} \right)} $.2) Show that for $s,\lambda >0$ and $0 \le...
  33. T

    Why do we use anti-derivatives to find the values of definite integrals?

    It seems like we calculate integration by doing the reverse of derivation. Differentiation is basically just using short-cuts for differentiation by first principles (e.g. power rule). If integration by first principles is the Riemann sum, then why don't we use short-cuts of the Riemann sum to...
  34. E

    Definite integrals: solving with residue theory and contour integration

    Homework Statement I need to solve this integral for a>0: \int _0^{\infty }\frac{\text{Sin}[x]}{x}\frac{1}{x^2+a^2}dx The Attempt at a Solution Using wolfram mathematica, I get that this integral is: \frac{\pi -e^{-x} \pi }{2x^2}=\frac{\pi (1-\text{Cosh}[a]+\text{Sinh}[a])}{2...
  35. B

    Questions about definite integrals

    While reading my calc book, I have developed a few questions about the situations in which definite integrals can exist. I've thought about these questions, and I feel that if I am able to answer some of them, I can make some other problems much easier, such as testing for convergence of a...
  36. D

    Derivative of the product of 2 definite integrals

    Homework Statement Find f'(x) for (integral from 0 to x of cos^5(t)dt)* (integral from x^2 to 1 of e^t^2 dt). No differentiation allowed in the answer Homework Equations The Attempt at a Solution. I used the product rule and integrated then differentiated the first term --> cos^5x*...
  37. D

    Can Definite Integrals Be Divided Algebraically?

    Is there a general algebraic way to write the quotient of two definite integrals as one? I mean, what would be \frac{\int_a^b f(s) ds}{\int_c^d g(t) dt} Is it analogous to the product of integrals creating a double integral? Thanks in advance!
  38. A

    Triangles and Definite Integrals

    I'm trying to figure out how to integrate a data set, without knowing the function. While doing this, I got to thinking about this: If the definite integral of a function can be represented by the area under that function, bound by the x axis, then shouldn't: \int_{a}^{b}2x\frac{\mathrm{d}...
  39. L

    Miscellaneous Definite Integrals

    Homework Statement show that \int^{∞}_{0}\frac{sin^{2}x}{x^{2}}dx= \frac{\pi}{2} Homework Equations consider \oint_{C}\frac{1-e^{i2z}}{z^{2}}dz where C is a semi circle of radius R, about 0,0 with an indent (another semi circle) excluding 0,0. The Attempt at a Solution...
  40. S

    Definite integrals with -infinity low bound

    I see equations of the form, y=\int_{-\infty }^{t}{F\left( x \right)}dx a lot in my texts. What exactly does it mean? From the looks of it, it just means there is effectively no lower bounds. I looked up improper integrals, but I can't say I really understand what is going on. So when...
  41. S

    MHB Can You Solve These Challenging Definite Integral Problems?

    Fun! Fun! Fun! Here are more entertaining problems: 1.\( \displaystyle \int_{2}^{4} \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(3+x)}+\sqrt{\ln(9-x)}}dx\) 2.\( \displaystyle \int_{\sqrt{\ln(2)}}^{\sqrt{\ln(3)}}\frac{x \sin^2(x)}{\sin(x^2)+\sin(\ln(6)-x^2)}dx\) 3.\( \displaystyle...
  42. C

    Solving Definite Integrals with Variable Limits

    Homework Statement I'm trying to solve a problem where you are asked to find the derivative of an integral where both limits of integrations contain variables Homework Equations Definition of integrals, derivatives The Attempt at a Solution For problems where you have an...
  43. A

    Definite Integrals Homework Solutions | Math Help

    Homework Statement I have to find the definite integrals of some problems... I did them in paint because i do not know how to do it on the forum so sorry if that is a problem Here is the problem and my attempted answer. I think i got the first one right, but the second one, someone told me...
  44. S

    Definite Integrals of Definite Integrals

    Homework Statement Suppose that f is a continuous function. a) If g is a differentiable function and if F(x) = \int_{0}^{x}g(x)f(t)dt find F ' (x). b) Show that \int_{0}^{x}(x - t)f(t)dt = \int_{0}^{x} [\int_{0}^{u}f(t)dt] du Suggestion: Use a) and the racetrack...
  45. Char. Limit

    Prove that two definite integrals are equal

    Homework Statement I want to prove the following statement: \int_0^{ln(2)} \sqrt{e^{2t} + 4 e^{4t} + 9 e^{6t}} dt = \int_1^2 \sqrt{1 + 4 t^2 + 9 t^4} dt Homework Equations The Attempt at a Solution To be honest, I'm not sure how to do this. I tried a substitution t=e^t for...
  46. J

    Definite integrals and areas under the curve

    Homework Statement Find the area under the cosine curve y=cosx from x=0 to x=b, where 0 is less than b is less than or equal to pi/2.Homework Equations \Sigmacos kx = [sin(1/2)(nx) cos(1/2)(n+1)x]/sin(1/2)xThe Attempt at a Solution I let n be a large integer and divided the interval [0,b] by n...
  47. J

    Basic Q on definite integrals and areas under the curve

    Homework Statement Consider the function f(x)=x2 on the interval [0,b]. Let n be a large positive integer equal to the number of rectangles that we will use to approximate the area under the curve f(x)=x2. If we divide the interval [0,b] by n equal subintervals by means of n-1 equally spaced...
  48. M

    I need a hint for this problem -Definite Integrals-

    Homework Statement Let F(x) = \int^{x}_{0}xe^{t^{2}}dt for x\in[0,1]. Find F''(x) for x in (0,1). Caution: F'(x)\neq xe^{x^{2}} Homework Equations The Attempt at a Solution I just need a hint. I know what F"(x) is already (solution was given), but I'd like to find F'(x) Thank you M
  49. M

    A very quick question about definite integrals

    Homework Statement F(x) = \int^{x}_{0}f(t)dt Then F'(x) = f(x) what is f'(x)? is this equivalent to f(t)? Thanks for your help M