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

    Evaluating definite integral by substitution

    Homework Statement By using substitution u=\frac{1}{t}, or otherwise, show that \int^∞_1 \frac{t^5}{(1+t^3)^3}dt=\int^1_0 \frac{u^2}{(1+u^3)^3}du Homework Equations The Attempt at a Solution Well, the reverse can also be done (making t to u). However, I don't know how to...
  2. J

    MHB Integral of $\frac{1}{(3+4\sin x)^2}dx$

    $\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$
  3. J

    Definite Integral of a Trig Fxn

    π/3 ∫tan^5 x sec x 0 I want to split the tan^5 x into a (sec^2 x - 1) but I only have sec x. So I'm a bit stuck. Help is appreciated.
  4. E

    How to evaluate the definite integral?

    How to evaluate the definite integral!? Homework Statement Integral test to see if the series converges or diverges. (infinity) Sigma ((n^2)/(e^(n/3)) (n=1) Homework Equations First I derive to make sure the function is decreasing and positive so it fits the...
  5. P

    Find the definite integral of a vector

    Homework Statement If \vec{r}(t) = t^2\vec{i} + t\cos{(\pi t)}\vec{j} + \sin{(\pi t)}\vec{k}, evaluate \int_{0}^{1} \vec{r}(t) \text{dt}. Homework Equations The Attempt at a Solution So I tried integrating each individual part, and I got \frac{1}{3}t^3\vec{i} + (-\pi t\sin{(\pi t)} -...
  6. mfig

    Transformation of variables in definite integral limits

    Hello, I am going through Whittaker's treatise on Classical Mechanics. In chapter 3 he derives the equation of motion for a simple pendulum, and I have a question about his method. Starting from the general form for the equation of energy (s is the path): \frac{m}{2}\dot{s}^2 =...
  7. S

    Evaluating a Definite Integral

    Homework Statement ∫ln(x+1)/(x+1)2dx from a=1, b=2 (Answer is -ln3/3+ln2/2+1/6) Homework Equations ∫fdg=fg-∫gdf The Attempt at a Solution Let u = (x+1), du=dx ∫lnu/u2du Let f=lnu , df = 1/udu, g=-1/u, dg=1/u2du =-lnu/u+∫1/u2du =[-lnu+1/u] 1->2 Using Fundamental Theorem...
  8. D

    Multiplication in a Definite Integral Equation

    If f(x)=\int_0^\infty g(x) dx and I wanted to multiply the integral by, say, a, what would I multiply the left side by? In other words, ? \times f(x) = a \int_0^\infty g(x) dx Thanks in advance!
  9. L

    Mastering Definite Integrals: Solving for the Square Root of 1+t^3 from 0 to 3

    Homework Statement Definite integral, from 0 to 3, of Square root of 1+t^3.Homework Equations Tried substitution The Attempt at a Solution There should be a simple way to do this but I can't seem to figure it out. Tried the substitution and whatnot but couldn't reach an answer.
  10. P

    Definite integral of exp and error function

    Hi, I've been trying to evaluate the following integral \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erf}\left(b\left(y-c\right)\right)\,\mathrm{d}y or equivalently \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erfc}\left(b\left(y-c\right)\right)\,\mathrm{d}y...
  11. B

    Proof for the limit of a definite integral where the integrand varies

    Homework Statement This problem was taken from Spivak's Calculus 3rd Edition, Problem 19-25 (d). Let f be integrable on [-1 , 1] and continuous at 0. Show that \lim_{h\rightarrow 0^+} {\int_{-1}^{1}{\frac{h} {h^2+x^2}}f(x)dx = {\pi}f(0).} Homework Equations I already proved from...
  12. T

    Definite integral evaluation (Inverse Fourier Sine Transform)

    Homework Statement I'm working on a long problem and have come to the final step. The answer seems so simple, but I can't quite get to it. I need to evaluate this integral: \int_0^{\infty}\ \left(e^{-k^2 t}/k\right)\sin(kx)\ dk Homework Equations Mathematica gives the result as...
  13. B

    MHB Definite Integral: x(-ln(x)^k)(1-x)^(N-1)

    Dear All, \int_{0}^{1}x*\left(\left(-ln(x)\right)^{k}\right)*\left(1-x\right)^{(N-1)}dx , where k is an odd no. N >=2.regards, Bincy
  14. B

    MHB Calculate Definite Integral with N Variable

    Hii friends,\int_{0}^{1}\left(1-x\right)^{N}*e^{x}\, dx regards, Bincy
  15. J

    How Do You Solve the Integral of (sin 37x / sin x) from 0 to π?

    Homework Statement Find the value of ∫ ( sin 37x / sin x )dx - from 0 to π Homework Equations The Attempt at a Solution Well i tried doing it by parts but it makes its more complicated. I tried to expand sin 37x as sin(36+1)x but could not solve it. I am lost
  16. S

    Arc length and definite integral

    Hi everyone ... I am a first time poster from South Africa. I have been visiting the forum for some time. I am busy teaching myself calculus and physics. I have hiccup with the concept of definite integral and arc length of a function. In my understanding these should be the same thing...
  17. M

    Absolute value definite integral

    Homework Statement ∫[|2x-1|] evaluated on the interval [1,-1] Homework Equations The Attempt at a Solution I know we must split it into two equations ∫ from [a,c] + ∫ from [c,b] and in absolute value one is negative and one is positive, so it will be 2x-1, and 1-2x my...
  18. P

    Definite Integral Calculator Problem

    Hi. Homework Statement I'm solving the definite integral 4π∫{0,3/2}(x^3)dx For some reason I get a different answer solving by hand and using my TI-89 than I do when I use an online calculator or my TI-36X Pro. I have to use the TI-36X Pro to take my Engineering FE exam in 2 & 1/2...
  19. N

    Find the limit a such that the definite integral equals 8

    Find the limit "a" such that the definite integral equals 8 Homework Statement Hello! the problem ask me to find for which a in R, does this holds: 2 ∫ abs(2x - x^2) = 8 a The Attempt at a Solution Well, I see that: f(x) { if x > 2, then f(x) is negative; if x < 2, then f(x) is positive...
  20. S

    Finding a power series expansion for a definite integral

    Homework Statement Find a power series expansion about x = 0 for the function f(x) = ^{1}_{0}\int\frac{1 - e^{-sx}}{s} ds Homework Equations The power series expansion for a function comes of the form f(x) = ^{\infty}_{0}\sum a_{k}x^{k} The Attempt at a Solution I've tried...
  21. P

    Integral Help: Solving ln(x) from 0 to 1

    the definite integral is: integral from 0 to 1 of ln(x). i used integration by parts (u=ln(x), du=1/x dx, dv=dx, v=x) to show that the integral is equal to: [x*ln(x)] (1,0) - integral from 0 to 1 of dx. this gives 1*ln(1)-0*ln(0)-(1-0) ln(1)=0, so the equation is now 0*ln(0)-1 0*ln(0) is an...
  22. Z

    Can Weierstrass Substitution Solve This Definite Integral?

    Hi everyone, when I was dealing with a physics problem, I find a definite integral and I cannot solve it. \int_{0}^{2\pi}\frac{1}{(a+cosx)^{2}}dx a is constant. Thank you!
  23. polygamma

    MHB Definite integral with four parameters

    Show that $\displaystyle \int_{0}^{\infty}\frac{x^{\alpha-1}}{(w+x^{\beta})^{\gamma}}\ dx = \frac{w^{\frac{\alpha}{\beta}-\gamma}}{\beta} B \left( \frac{\alpha}{\beta}, \gamma - \frac{\alpha}{\beta} \right) = \frac{w^{\frac{\alpha}{\beta}-\gamma}}{\beta} \frac{\Gamma \left(...
  24. N

    MHB U-substitution, definite integral, trig function

    problem: sin(x)^2cos(x) integrate from 0 to pi/2the way I tried this is by making u = sin, and then working with du to make cos cancel out the next thing I did was check my new intervals by plugging in pi/2 and 0 into the u function (becomes 0 to 1) next, I took the anti derivative of u, which...
  25. J

    U substitution with definite integral

    Homework Statement make a u- substitution and integrate from u(a) to u(b) Homework Equations ∫[0,1] √(t^5+2t) (5t^4+2) dt The Attempt at a Solution u= t^5+2t du= 5t^4+2 u(1)=2 u(0)=0 ∫[0,1] √(u) du (2/3)(u)^3/2+c l[0,2] (2/3)( 0^5+2(0))^3/2- (2/3)(2^5+2(2))^3/2 0+...
  26. J

    Definite Integral of Trig Function

    Homework Statement ∫sin2xcosx x=0,pi/4 Homework Equations The Attempt at a Solution By double angle magic: ∫sin2xcosx= ∫2sinxcos^2x u=cosx du=-sinxdx -2∫u^2 -2(u^3)/3 -2cos^3x/3 = -1/(3√2) for x=pi/4 How does that look folks?
  27. T

    Having problems evaluating definite integral

    Homework Statement I'm trying to evaluate \int_0^{2\pi} \frac{9-6\cos t}{5-4\cos t} dt \ . I found that the antiderivative of the integrand is F(x) = \frac{3}{2}x + \tan^{-1} \left(3\tan\frac{x}{2} \right) so using the FTC, the integral should be F(2\pi)-F(0)=3\pi But using a...
  28. Calculuser

    I hope that helps! Proof of Definite Integral??

    Proof of Definite Integral?? Riemann Sum helps us to find the are under a curve.But I couldn't understand connection between Riemann Sum and Definite Integral.I mean; \int^{b}_{a}f(x).dx=F(b)-F(a) How can I prove F(b)-F(a) in definite integral?? Thanks..
  29. B

    Substitution of a definite integral

    Homework Statement The Attempt at a Solution I don't see why the √v disappears in this step I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^1/2)/(1/2)
  30. B

    Substitution of a definite integral

    Homework Statement Why does the 3x^2 disappear? Doesn't it play a role in the answer?
  31. B

    Express sum as a definite integral

    Homework Statement If n is a positive integer, then \lim_{n\to\infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^2+\left(\frac{2}{n}\right)^2+\cdot\cdot\cdot+\left(\frac{n-1}{n}\right)^2\right]can be expressed by what definite integral? The attempt at a solution A student I was helping...
  32. N

    Evaluate the definite integral.

    Homework Statement Integrate from 0 to 1 x47e-x48 Homework Equations The Attempt at a Solution u = -x48 du = -48x47dx dx = du / -48x47 ∫eu * du/-48 -\frac{1}{48} ∫ eu du -\frac{1}{48} eu -\frac{1}{48} e1 - ( -\frac{1}{48} e0) -\frac{e}{48} +...
  33. S

    MHB Can You Solve This Complex Trigonometric Integral?

    Can you solve the Integral? \[ I(a,b)=\int_{0}^{\pi / 2}\frac{1}{(a\cdot\cos^2{x}+b\cdot\sin^2{x})^2}dx \quad a,b>0 \]
  34. S

    Definite Integral of Both Sides

    Homework Statement "Integrate both sides of these equations from some starting point to some ending point (such as t0 to t): r dt = d[A], C dt = dH and dw = (nRT/V) dV Homework Equations -- The Attempt at a Solution I'm have trouble understanding the concept of taking the...
  35. C

    How Is the Definite Integral of a Trigonometric Function Evaluated Correctly?

    Homework Statement \int\stackrel{tan(8)}{tan(1)}\frac{dx}{x^{2}+1} Homework Equations The Attempt at a Solution \int\stackrel{tan(8)}{tan(1)}\frac{dx}{x^{2}+1}tan^{-1}x|\stackrel{tan(8)}{tan(1)}tan^{-1}(tan(8))-tan^{-1}(tan(1))8-1=7I'm fine with this, but my instructor said "Hint: The answer...
  36. D

    Difficult Definite Integral Question

    1. Find the expected value of the largest order statistic in a random sample of size 4 from the standard normal distribution. 2. Homework Equations E(X(4,4))=4∫xf(x)(F(x))^3dx, (from minus infinite to plus infinite), where f(x) is the probability density function of standard normal...
  37. L

    Solving Definite Integral Problem: ∫3,6(2f(x)−3)dx=12

    We are just starting integrals right now and I'm having trouble with this problem. I know how to solve a definite integral but I don't quite understand what is being asked here? Just wondering if anyone could let me know where I should start? if∫0,3f(x)dx=12, ∫0,6f(x)dx=42,find...
  38. H

    Definite Integral of the Natural Log of a Quadratic using Riemann Sums

    Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...
  39. E

    Evaluate Definite Integral Using Trapezium Rule: 4, 8 & 16 Subs

    ∫cos x + sin x lower limit: 0 upper limit: pi correct answer: 4 intervals: 1.8962, 8 intervals: 1.9742, 16 intervals: 1.9936 i cannot seem to get the correct answer, here is my attempt: x values cosx + sinx values 0...
  40. J

    Simplify definite integral formula

    Homework Statement Working my way through Zienkiewicz's "FEM its basis & fundamentals" I am stuck on the following example: - \int_{y_1}^{y_2} t N1 \left\{ \begin{array}{c} \sigma_x \\ \tau_y \end{array} \right\} dy = -\left\{ \begin{array}{c} k_x t (2 y_1 + y_2) (y_2 - y_1) / 6 \\ 0...
  41. A

    Proving Equality of Definite Integrals with Trigonometric Functions

    Homework Statement I've been solving a problem, the solution is complete, however, I must prove that the following relation is true: \int_{0}^{\frac{\pi}{2}} sin^m 2x dx = \int_{0}^{\frac{\pi}{2}} cos^m x dx for any m. Homework Equations - The Attempt at a Solution Well, I've trying to...
  42. P

    Can someone check me on this definite integral

    Homework Statement integral from 0 to pi of (x sinx)/(1+cos^2x)Homework Equations I just need to know if I did it rightThe Attempt at a Solution I got (5pi/6)-1/9
  43. S

    Need helping evaluating this limit by expressing it as a definite integral

    Homework Statement Evaluate the limit by expressing it as a definite interval integral \frac{lim}{n \rightarrow \infty} (\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+(n-1)}+\frac{1}{n+n} Homework Equations The Attempt at a Solution OK, so I know it has to do with Rieman sums...
  44. S

    Definite integral using Riemann sums?

    I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...
  45. R

    Help Calculating Definite Integral

    Homework Statement (2(x)^2)/((x+1)((x)^2+1)) from 0 to 1 The indefinite answer is (1/2)ln2+ln2-(pi/4) How did it get to this answer?(3/2)ln2-(pi/4)
  46. S

    Definite integral of complex gaussial-like function

    so the question is how to solve the next: ∫...∫exp(-(a+b*x1+c*x1*x4+d*x3*x4+f*x2*x3*x4)^2)dx1dx2dx3dx4 a,b,c,d,f - real constants; x1,x2,x3,x4 - real variables; borders of integration are finite, let say for x1 it is [x1a,x1b] and so on for the rest of them; i'v tried to solve it in...
  47. J

    Definite integral using complex analysis

    Homework Statement Considering the appropriate complex integral along a semi-circular contour on the upper half plan of z, show that \int^{\infty}_{\infty} \frac{cos(ax)}{x^2 + b^2} dx = \frac{\pi}{b}e^{-ab} (a>0, b>0) Homework Equations \int_{C} = 0 For C is a semi-circle of...
  48. L

    Evaluate a fraction in a definite integral?

    I don't understand how to evaluate a fraction in a definite integral, could anyone please explain this problem to me? int. between 3 (lower integral) and 4 (upper integral) of 3x^2+2/x^2 I thought of using the substitution law, and got as far as 1/6 u * du/x ...I don't quite know how to...
  49. D

    Definite integral with square on bottom

    Homework Statement \int_{1/2}^{\sqrt{3}/2}\frac{6}{\sqrt{1-t^2}}dt Homework Equations The Attempt at a Solution Ok, this is an odd problem I'm working from the book. The book says the answer is \pi. First I tried getting rid of the fraction: \frac{6}{\sqrt{1-t^2}}dt =...
  50. V

    Definite integral (e^x) *(x-1)^n=16-6e find n.

    Homework Statement this is definite integral question. lower limit 0 ; upper limit 1 ; integral (e^x)(x-1)^n = 16-6e find n (n<6) here e is euler constant value around 2.7 (irrational) hope you understood. Homework Equations as much as first year student know. The Attempt at a Solution...
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