Integrals Definition and 1000 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. M

    (Iterated Integrals) Volume between a Cone and a Sphere

    Homework Statement This is a book problem, as follows: Find the volume between the cone x = \sqrt{y^{2}+x^{2}} and the sphere x^{2}+y^{2}+z^{2} = 4 Homework Equations spherical coordinates: p^{2}=x^{2}+y^{2}+z^{2} \phi = angle from Z axis (as I understand it) \theta = angle from x or...
  2. W

    Trigonometric Integrals Technique

    Homework Statement When integrating a function of the form: \displaystyle\int_c^d { (sinx)^{a} * (cosx)^{b}} Is this a correct simplification of the rules to evaluate: 1. if exponent on sin or cos is odd, and the other is even, separate out one of the odd's and use an identity on the...
  3. M

    Trig integrals and finding volume.

    Homework Statement Find the volume obtained by rotating the following curves bounded by: y=sinx y=0 pi/2≤ x ≤ pi Homework Equations I know I have to use the cylindrial disk method so ∫2pi(x)f(x)dx. The Attempt at a Solution I did the following ∫(pi/2 to pi) 2pi (x) sinxdx...
  4. T

    Surface integrals and parametrization

    An area A in the xy-plane is defined by the y-axis and by the parabola with the equation x=6-y^2. Furthermore a surface S is given by that part of the graph for the function h(x,y)=6-x-y^2 that satisfies x>=0 and z>=0. I have to parametrisize A and S. Could this be a...
  5. M

    Improper Integrals, Infinite Limits

    Homework Statement ∫e-Sxsin(ax) dx, S and A are constants, upper limit is ∞ lower is 0 Homework Equations ∫ u dv = uv - ∫ vdu The Attempt at a Solution After integrating by parts twice I got: (S2)/S(S2+a2) lim c→∞ [-sin(ax)e-Sx + acos(ax)e-Sx] |^{C}_{0} Okay, now how on Earth do I take...
  6. E

    Transformations of Double Integrals with Rectangular Domains in the 1st Quadrant

    Suppose we have the double integral of a function f(x,y) with domain of integration being some rectangular region in the 1st quadrant: 0≤a≤x≤b, 0≤c≤y≤d. Would the following transformation generally be acceptable? (I've quickly tried it out several times with arbitrary integrands and domains...
  7. T

    Integrals: #1 Help with fraction #2 Moment of inertia

    Homework Statement 25-2-EX9 The time rate of change of the displacement (velocity) of a robot arm is ds/dt = 8t/(t^2 + 4)^2. Find the expression for the displacement as a function of time if s = -1 m when t = 0 s. 26-5-9 Find the moment of inertia of a plate covering the first-quadrant region...
  8. Mandelbroth

    Derangements and Contour Integrals?

    I did a proof a few days ago (for the sake of enjoyment) and my teacher thought it was interesting, though he seemed unsure of my result. Consider a set of n distinct objects, P. If n \in \mathbb{Z}_+ \cup \left\{0\right\}, then the cardinality, q, of the set of all derangements of P is...
  9. M

    Symbol for partial derivative not used for partial integrals?

    {\frac{∂(xy)}{∂x}=x} Going backwards. If we took, ∫x dy we get xy+f(x) Now, the only way that ∫x dy is a valid operation, is if we know that we came from a partial derivative. Why, when taking a partial...
  10. A

    Substitution formula for integrals

    I suppose you all know the substitution formula for integrals. Well sometimes it seems to me you use substitutions which just don't fit directly into that formula. For instance for the integral of 1/(1+x^2) you substitute x=tan(u). Why is it suddenly allowed to assume that x can be...
  11. C

    Help understanding closed line integrals

    Hi I'm currently studying Electromagnetism, and we keep coming across this symbol: \oint A closed line integral, something I have never really been able to understand. If a normal integral works like this: http://imageshack.us/a/img109/3732/standardintegral.png where f(x) is the "height"...
  12. T

    Double Integrals in polar coordinates: Calculus 3

    Homework Statement Given \int^{\sqrt{6}}_{0}\int^{x}_{-x}dydx, convert to ploar coordinates and evaluate. Homework Equations We know that x=rcos\theta and y=rsin\theta and r =x^2+y^2 The Attempt at a Solution First, I defined the region of the original integral: R = 0...
  13. R

    How do you compute the circulation of this fluid (path integrals)

    Homework Statement A fluid as velocity field F(x, y, z) = (xy, yz, xz). Let C denote the unit circle in the xy-plane. Compute the circulation, and interpret your answer.Homework Equations The Attempt at a Solution Since the unit circle is a closed loop, I assumed that ∫ F * dr = 0 (the ∫ symbol...
  14. B

    Physics applications of integrals

    Homework Statement a region that resists a body of water follows the curve y=.3x^2 from 0<=y<=240 using water density of 1000 kg/m^3 and g of 9.8 m/sec^2 Homework Equations 0 to 240∫g(rho)(240-y)2(sqrt(y/.3) The Attempt at a Solution 9.8(1000)(2)(240^(3/2)(292.199-.730297(240)) i...
  15. C

    Maximizing Integration Efficiency: Long Division vs Partial Fractions

    Homework Statement ∫(2x+1)/(x²+2x+1)(x²+x+1) Homework Equations none The Attempt at a Solution I've foiled this out to look like: ∫(2x+1)/(x^4+3x³+4x²+3x+1) I'm trying long division here but it's getting really ugly really fast. Should I foil this out in the first place or...
  16. E

    Closed form solutions to integrals of the following type?

    For any integral where the integrand is of the form f(θ)^z, with z a complex number, and f(θ) = sin(θ), cos(θ), tan(θ), ... etc., θ being either real or complex. Is it possible to explicitly solve for an antiderivative? I'm not aware of any such way I could use residues/series representations...
  17. K

    How to Solve Integrals Using Trig Substitution?

    Homework Statement integral of dx/((9-(x^2))^(3/2)) A = 0, B = 3/2 Homework Equations Trigonometry Substitutions 3. The Attempt at a Solution : I am stuck with this question. So far, I got (1/9)integral of (1/cos^2(θ)) dθ
  18. R

    Path integrals and parameterization

    Homework Statement Evaluate ∫ F ds over the curve C for: a) F = (x, -y) and r(t) = (cos t, sin t), 0 ≤ t ≤ 2∏ b) F = (yz, xz, xy) where the curve C consists of straight-line segments joining (1, 0, 0) to (0, 1, 0) to (0, 0, 1) Homework Equations The Attempt at a Solution a) I first found the...
  19. M

    Curl and its relation to line integrals

    hey all i know and understand the component of curl/line integral relation as: curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr where we have vector field F, A(C) is the area of a closed boundary, u is an arbitrary unit vector, dr is an infinitely small piece of curve C my...
  20. M

    Find center of mass and coordinates using double integrals?

    Homework Statement Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is bounded by the parabolas y = x^2 and x = y^2; ρ(x, y) = 23√x Homework Equations m = \int\int_{D} ρ(x, y) dA x-bar = \int\int_{D} x*ρ(x, y) dA y-bar =...
  21. phosgene

    Improper integrals and solids of revolution

    Homework Statement Let n>1/2 and consider the function f(x)=x^{-n} for x\in[1,∞) Calculate the volume of the solid generated by rorating f(x) about the x-axis, showing all details of your working. Homework Equations Since it is rotated about the x-axis, its axis of symmetry is...
  22. F

    Area of an ellipse using double integrals

    I can do this calculation using different methods; my interest is improving my skills at using this method, rather than the answer. Trying to find the area of the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 From the Jacobian, we get dxdy = rdrd\theta So I go from the above equation of the...
  23. Fernando Revilla

    MHB Fresnel Integrals: Unsolved Question from MHF

    I quote a unsolved question posted in MHF by user poorbutttryagin on February 5th, 2013.
  24. STEMucator

    Setting up some triple integrals

    Homework Statement I want to know if I've gone about setting up these integrals in these questions properly before I evaluate them. (i). Find the mass of the cylinder S: 0 ≤ z ≤ h, x^2 + y^2 ≤ a^2 if the density at the point (x,y,z) is δ = 5z^4 + 6(x^2 - y^2)^2. (ii). Evaluate the...
  25. E

    More general formula for integrals

    I was wondering: Is there an even more general formula for the integral than int(x^k) = (x^(k+1))/(k+1) that accounts for special cases like int(x^(-1)) = ln|x| and possibly u substitutions?
  26. A

    Trigonometric Integrals by Substitution

    I am unsure whether I have properly performed the integration of the integral ∫((sin(√x))^3*dx)/√x When I used my TI-Nsprire CAS to take the derivative of my answer in order to check if I was correct, and it came out differently. Now I used some trig identities to manipulate the problem, so I...
  27. D

    Integrals and Convolution: How to Group Multiple Functions Together?

    Hi, I shall show the following: (f*g) \star (f*g) = (f\star f)*(g\star g) where * denotes convolution and \star cross-correlation. Writing this in terms of integral & regrouping: \int_{\phi} \left(\int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1\right) \cdot \left(\int_{\tau_2} f(\tau_2)...
  28. P

    Are the following two integrals equivalent? [basic]

    Hi, this is rather short one. I'm wondering if the two in the image below are equivalent. http://gyazo.com/62449de717e91b412f42771fd7d0a1af To me they appear to be equivalent, and I can't recognize any exceptions or trivial cases. Also, if it is equivalent would be it be acceptable to...
  29. S

    The mean value theorem for integrals and Maple

    Homework Statement I have function f which is defined upon an interval [a,b]. I have calculated the mean value using the theorem \frac{1}{b-a} \int_{a}^b f(x) dx What I would like to do is to plot in Maple the mean value rectangle. Where the hight of this rectangle represents the mean value...
  30. S

    Thorycal Issue with Improper Integrals

    So, like i said in the Title this more of a theorycal question. In my university notebook i have written that an integral to converge has to happen the next: 1. The f has to be bounded (if not its just a dot) 2.The interval has to be finit. [THIS IS WHAT IT'S WRITTEN IN MY NOTEBOOK] See, my...
  31. A

    Using double integrals to evaluate single integrals

    In most calculus textbooks, they use double integrals to evaluate the Gaussian integral. Where did they get the idea - or how did they choose the two variable function e^{-(x^2+y^x)} to evaluate it? I guess this is related...but if you were given a fairly hairy integral and it was suggested...
  32. A

    Triple integrals, changing the order of integration

    Homework Statement Write out the triple integral for the volume of the solid shown in all six possible orders. Evaluate at least 2 of these integrals. Homework Equations I attached a picture of the figure. The front : x/2+z/5=1...
  33. W

    Disk, Washer, Shell Multiple Integrals

    Homework Statement Determine how many integrals are required for disk, washer, and shell method.Homework Equations x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1) about x-axis.The Attempt at a Solution Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for...
  34. S

    I don't understand how to differentiate integrals

    I am in Calculus 2 and we're just reviewing calc 1. Can someone break down the concept of differentiating definite integrals for me? I am mostly struggling on the trig functions. The problem I am stuck on is ∫^{}x_{}0 cos(t^{}2) dt
  35. L

    Transforming double integrals into Polar coordinates

    Homework Statement Show that: I = \int\int_{T}\frac{1}{(1 + x^{2})(1 + y^{2})}dxdy = \int^{1}_{0}\frac{arctan(x)}{(1 + x^{2})}dx = \frac{\pi^{2}}{32} where T is the triangle with successive vertices (0,0), (1,0), (1,1). *By transforming to polar coordinates (r,θ) show that:* I =...
  36. 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...
  37. L

    Finding volumes via double integrals

    Homework Statement Find the volume which lies below the plane z = 2x + 3y and whose base in the x - y plane is bounded by the x- and y-axes and the line x + y = 1. Homework Equations I = \int\int_{R} f(x, y) dydx = \int^{b}_{a}\int^{y=y_{2}(x)}_{y=y_{1}(x)} f(x, y) dydx The...
  38. I

    Calculus 2: Finding Work with Integrals

    Homework Statement Not sure if this goes here or physics, but this is for my calculus 2 class so I decided here would be best. #22 Homework Equations W=∫Fdx The Attempt at a Solution I think the limits of integration are from 0 to 1 since the water is right under the spout, but...
  39. C

    Calc II homework - substitution of definite and indefinite integrals

    It's been a year since I took Calc I, and I'm taking Calc II online this semester. This is technically a review problem from Calc I, and I managed the other seven, but I can't figure out how to solve this problem. 1.a Homework Statement ∫(a*sin(14x))/(\sqrt{1-196x^2} dx, evaluated at x=0...
  40. K

    How to Calculate Gaussian Integrals in QFT?

    1. Gaussian Please help me calculate some Gaussian integrals in the attached file which are used in my QFT calculations. Thank you very much!
  41. K

    How to Calculate Gaussian Integrals with 4-Momentum in QFT?

    I am doing some calculations in QFT. And, in my calculations, I have to deal with 5 Gaussian integrals as followed. Please help me calculate those 5 integrals. Thank you very much!
  42. T

    Differential Spherical Shells - Triple Integrals

    Homework Statement Despite the fact that this started as an extended AP Physics C problem, I turned it into a calc problem because I (sort of) can. If it needs to be moved please do so. There is a hollow solid sphere with inner radius b, outer radius a, and mass M. A particle of mass m...
  43. P

    MHB Newton's method to approximate integrals?

    Can we use Newton's method to approximate the value of definite integrals? (Thinking) EDIT: Ignore if the question doesn't make sense (which it probably doesn't).
  44. T

    (Improper Integrals) Not sure if I'm doing this properly

    Initial improper integral: ∫ dx / (1+x**2) * (1+ atan(x)) , x = 0, ∞ Substitutions: μ = 1 + atan(x) dμ = dx / (1 + x**2) μ(∞) = 1 + pi/2 μ(0) = 1 Integral: ∫ dμ / μ , μ = 1, 1+ pi/2 Then solve. I'm getting the right answer, but I think I'm botching something due to a lack of...
  45. O

    Double Integrals, Rectangular Region

    Homework Statement Using ∫∫kdA = k(b-a)(d-c), where f is a constant function f(x,y) = k and R = [a,b]x[c,d], show that 0 ≤ ∫∫sin∏xcos∏ydA ≤ 1/32, where R = [0,1/4]x[1/4,1/2]. Homework Equations ∫∫kdA = k(b-a)(d-c) 0 ≤ ∫∫sin∏xcos∏ydA ≤ 1/32 The Attempt at a Solution I tried to...
  46. O

    Multiple integrals: Find the volume bounded by the following surfaces

    Homework Statement Find the volume bounded by the following surfaces: z = 0 (plane) x = 0 (plane) y = 2x (plane) y = 14 (plane) z = 10x^2 + 4y^2 (paraboloid) Homework Equations The above.The Attempt at a Solution I think it has something to do with triple integrals? But...
  47. T

    Connection between definite and indefinite integrals?

    I understand that the indefinite integral is like infinite definite integrals, but how come when we calculate the definite integral we simply substitute the two values into the indefinite integral and subtract? Why do we subtract? Why not add? Also, there aren't the same thing, right? What's...
  48. MikeGomez

    Simple integrals for gravitational potential

    Homework Statement Homework Equations I need help solving intergral… \int \frac{dx}{(a+x)^2} The Attempt at a Solution I found the integral for… \int \frac{dx}{(a^2+x^2)} = 1/a arctan x/a But I don’t know how to apply that to the original integral which is a little different...
  49. D

    2D Systems and 4D Minkowski Space: Exploring Path Integrals

    The combination of special relativity and quantum mechanics in a single framework makes our understanding of such systems to be true only in 4D, Minkowski space...I have noticed that recent published work concerning 2D systems and I am not sure about this reduction of 4D to only 2D, does it mean...
  50. X

    Simple volume calculation problem (double integrals)

    [EDIT]: Found the mistake, see the next post. Homework Statement Evaluate $$\iint_{S}{\rm e}^{x+y}dx\, dy,S=\{(x,y):\left|x\right|+\left|y\right|\leq1\} $$ 2. The attempt at a solution ##\left|x\right|+\left|y\right|## is the rhombus with the center at the origin, symmetrical about both...
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