What is Double integral: Definition and 573 Discussions

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in





R


2




{\displaystyle \mathbb {R} ^{2}}
(the real-number plane) are called double integrals, and integrals of a function of three variables over a region in





R


3




{\displaystyle \mathbb {R} ^{3}}
(real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

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  1. I

    How to Approach a Complex Double Integral with Trigonometric Functions?

    [SOLVED] Need Help With a Double Integral Any help on the following integral would be appreciated. I don't know where to begin at all. \int_0^{1}\int_{arcsiny}^{\pi/2} sec^2(cosx) dxdy I've thought about changing the order of integration, but I don't think that will...
  2. tony873004

    Evaluate Double integral

    Homework Statement Evaluate the integral \int {\int\limits_R {\left( {x + y} \right)\,dA} } where R is the region that lies to the left of the y-axis between the circles x^2 + y^2 = 1 and x^2 + y^2 = 4 by changing to polar coordinates. Homework Equations x=r cos theta y=r sin...
  3. R

    Double Integral of two concentric circles

    Homework Statement Let D be the region given as the set of (x,y) where 1 <! x^2+y^2 <! 2 and y !<0. Is D an elementary region? Evaluate \int\int_{D} f(x,y) dA where f(x,y) = 1+xy. Homework Equations The Attempt at a Solution So I understand that this is two concentric circles(an...
  4. S

    Find the Volume (Double Integral)

    I'm having trouble trying to setup this double integral. The question asks to find the volume of a solid enclosed by the parabolic cylinder y = x^{2} and the planes z = 3y, z = 2+y I'm not even sure where to start. I have drawn the figure and understand that you have to integrate the two...
  5. B

    Double integral with cos(x^n) term

    [SOLVED] Double integral with cos(x^n) term Homework Statement Solve the following integral (without using a series development): \displaystyle \int _{0}^{\frac{1}{8}}\int _{\sqrt[3]{y}}^{\frac{1}{2}}\cos\left(20{\pi}x}} ^{4}\right)dx dy Homework Equations N/A The Attempt at a...
  6. G

    Evaluate Double Integral Problem

    Homework Statement Evaluate \int\int e^x^2 dx dy. The bounds for the inner integral go from y to 1 The bounds for the outer integral go from 0 to 1 2. The attempt at a solution I can easily do this, I just do not see how I can get e^x^2 to integrate for x. Is there some sort of special...
  7. U

    Double Integral Help - Problem 105 | UFL Math

    I can do this problem if they give a general equation, but the infinite boundaries are confusing me. I don't know how to inert all the symbols so I'll link you to the problem. It's problem number 105 in the packet. http://www.math.ufl.edu/%7Ehuang/calc3/fall2007.pdf
  8. B

    Efficiently Solve a Tricky Double Integral with These Proven Methods

    Homework Statement Evaluate: \int_{0}^{4} \int_{\sqrt{x}}^{2}e^y^3dxdy The Attempt at a Solution Well that's a Fresnel type function so you can't find an antiderivative for it. I'm pretty sure the point of this assignment isn't Taylor series so I'm quite certain we aren't...
  9. N

    Transforming a Double Integral with Constant Function Along Parallel Lines

    I'm supposed to prove that \int\int_{S}^{}\ f(ax + by + c) \, dA \ =2 \int_{-1}^{1} \sqrt{1 - u^2} f(u\sqrt{a^2 + b^2} + c) \, du Where S is the disk x^2 + y^2 <= 1. It is also given that a^2 + b^2 is not zero I can´t use polar coordinates and I can´t see how you simplify the surface S...
  10. N

    Interpreting a double integral as volume

    The function F(x,y) = 4x^2y^3 over the disk x^2 + y^2 =1 is supposed to be zero over the disk. I'm wondering how you can see it? I cannot see this or imagine it in 3D. Is it because the function is odd in terms of y? F(x,-y) = -F(x,y) ? independent of wheher x is positive or negative...
  11. G

    Substitution with double integral

    Homework Statement Using transforms: u = 3x + 2y and v = x+4y solve: \iint_\textrm{R}(3x^2 + 14xy +8y^2)\,dx\,dy For the region R in the first quadrant bounded by the lines: y = -(3/2)x +1 y = -(3/2)x +3 y = -(1/2)x y = -(1/2)x +1 I'm itching to see where I've gone wrong on this one...
  12. A

    Absolute Value in a double integral

    [SOLVED] Absolute Value in a double integral Homework Statement If \Omega = [-1,1] x [0,2], evaluate the double integral \int\int_{\Omega} \sqrt{|y-x^{2}|} dA given that it exists. Homework Equations None The Attempt at a Solution I know that in order to integrate with the...
  13. P

    Evaluating a Double Integral Using Polar Coordinates

    Okay I have no idea where to start on this example problem: Use polar coordinates to evaulate the double integral e^((x^2)+(y^2))dydx [frist (inner) integal lower limit y= -sqrt(4-x^2) upper limit y=0)] [second (outer) lower limit x=0 upper limit x=2] When I start doing the integral...
  14. B

    Integrating this double integral

    Hi, I am having some difficulties integrating the following expression.. \int\int\left(\frac{k}{\pi}\right)^2 \frac{1}{k^2+\omega'^2}\frac{1}{k^2+\omega^2}e^{-i\omega'\tau}e^{i\omega\tau}d\omega d\omega' I've tried by part but it doesn't look like it's going to give me the right answer...
  15. E

    Double integral question: Evaluating Integrals with Sinusoidal Functions

    Double integral question... Homework Statement Evaluate the integrals shown ( I have attached the file with the integral). Homework Equations The Attempt at a Solution Ok, for the first one, can you tell me how I integrate sin x^2...?? which method should i use? And for the...
  16. E

    Double integral problem help appreciated

    Homework Statement Evaluate the integral shown ( I have the file with the given integral attached here). Homework Equations The Attempt at a Solution So what i did was change dy dx into dx dy. Then i integrated y so the whole thing becomes 2x - y^3. I plugged the values (1+x)...
  17. R

    Double Integration with Polar Coordinates

    1. Integrate f(x,y)=x+y 1<=x^2+y^2<=4, x>=0, y>=0 3. ∬x+y dxdy x=rcos(o) y=rsin(o) ∬r(rcos(o)+rsin(o))drdo r is from 1 to 4, o is from 0 to pi/2 I get the wrong answer and don't know why
  18. N

    Double integral into the polar form

    hello i have this problem about polar form, i am aware that when you have a problem like \int\int x^2 + y^2 dxdy you use r^2 = x^2 + y^2 but i what would you do if you had a problem like \int\int xy dxdy? thanks in advance. edit: i know the limits if you need them please let me know but i...
  19. R

    Supposedly simple double integral

    double integral of xy dA in the triangular region of (0,0), (3,0), (0,1). my problem that I am having is finding the limits I am suposed to find dx or dy in. I figure I should use 0 to 3 for dx, but then i do dy from 0 to what? Help appreciated.
  20. E

    How to Change the Order of Integration in a Double Integral?

    Homework Statement Evaluate the integral shown in the diagram Homework Equations The Attempt at a Solution The first step to evaluating the integral is shown in the diagram (labelled as 2). They said they changed the order of integration. I was wondering what they mean by...
  21. B

    Double integral of mass of circular cone

    Find the mass of a right circular cone of base radius r and height h given that the density varies directly with the distance from the vertex does this mean that density function = K sqrt (x^2 + y^2 + z^2) ? is it a triple integral problem?
  22. B

    Double integral of volume bounded by plane and paraboloid

    Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2 I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.
  23. S

    Troubleshooting Flux Out of a Cube: Evaluating a Double Integral

    I am trying to work through some examples we have been given on flux out of a cube but am having difficulty in seeing how one one line of the answer becomes the next. The question is analysing the flux out of a cube by looking at each side individually and working out the surface integrals...
  24. M

    Double integral coordinate transform

    Basically I want to find the new limits w,x,y,z when we make the valid transformation \int^{\infty}_0 \int^{\infty}_0 f(t_1,t_2) dt_1 dt_2 = \int^w_x \int^y_z f(st, s(1-t)) s dt ds I've tried putting in arbitrary functions f, and so getting 4 equations constraining the limits, but I end up...
  25. E

    Is Boundedness a Necessity for Double Integral Proofs?

    Show that if f is defined on a rectangle R and double integral of f on R exists, then f is necessarily bounded on R.
  26. K

    Can a Double Integral be Simplified Using a Substitution of Polar Coordinates?

    If we wish to calculate the integral. \int_{0}^{\infty}dx \int_{0}^{\infty}dy e^{i(x^{2}-y^{2}} which under the symmetry (x,y) \rightarrow (y,x) it gives you the complex conjugate counterpart. my idea is to make the substitution (as an analogy of Laplace method) x=rcosh(u) ...
  27. E

    Double Integral Help: |cos(x+y)| over [0,pi]x[0,pi]

    doubleIntegral( |cos(x+y)| dx dy ) over the rectangle [0, pi]x[0,pi] I tried several ways to split the integral up so that I could remove the absolute value sign and integrate. However, I did not get the correct answer, so I must be splitting it wrong. Can someone show me how to split the...
  28. W

    When does this double integral converge?

    For the double integral find which values of k make it converge. \int \int \frac{dA}{(x ^ 2+y^2)^k} x^2 + y^2 <= 1 I have no idea how to even start going about this, can just about do the basics of multiple integration but not this.
  29. K

    Solving double integral without integrating

    From an example in my book: Int Int (2x) dxdy over R = 0 (R is the circe x^2+(y-1)^2=1) How can one make this conclusion without integrating?
  30. K

    Double Integral of x*y^3 + 1 over Surface r=1, tetha 0-Pi, z 0-2: Solving Guide

    Int Int (x*y^3 + 1) dS where S is the surface r=1, tetha from 0 to Pi and z from 0 to 2. How can I solve this integral? I haven't got a clue.
  31. C

    Need some double integral help.

    Evaluate. double integral (e^(y^3)) dy dx Where dy is evaluated from sqrt(x/3) to 1 ...and dx is evaluated from 0 to 3. I am lost. I don't even know how to start.:frown:
  32. lemma28

    Help with double integral of exp(ixy)

    Please help me with folllowing double integral \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {e^{ixy} dxdy = 2\pi}} (x,y, real) It came out analyzing the relation between DiracDelta and the Fourier Transform formula. (it's the reason why insert the constant...
  33. quasar987

    Double Integral: Finding Area of Paraboloid Beneath z=2

    It's about finding the area of the paraboloid z=x²+y² beneath z=2. The area integral is \int\int_{\{(u,v): u^2+v^2<2\}}\sqrt{1+4(u^2+v^2)}dudv A polar change of variable seems to fits nicely: =\int_0^{\sqrt{2}}\int_0^{2\pi}\sqrt{1+4r^2}rd\theta dr Then the change of variable \xi=1+4r^2...
  34. G01

    Is this double integral set up correctly?

    1.Set up the integral to Find the volume enclosed by the cylinder x^2 +y^2 = 1 x=0 and z=y 3. The area to integrate over is the part of x^2 + y^2 =1 above the x axis. X goes from -1 to 1 and y goes from 0 to sqrt(1-x^2) So the integral should be: \int^1_{-1} \int^{\sqrt(1-x^2)}_0 y dy dx
  35. G01

    How can we use a polar double integral to derive the volume of a sphere?

    Hey everyone, My task this time is to derive the volume of a sphere using a polar double integral. The sphere has radius a we know that r goes from 0 to a in this integral. The equation for a sphere is: x^2 + y^2 +z^2 = r^2 or f(x,y) = \sqrt{r^2 -x^2 -y^2} and it intersects the x-y plane...
  36. O

    Finding Moment of Inertia of Infinitely Thin Hoop using a Double Integral

    Here is the problem: http://img141.imageshack.us/img141/3830/problemsm5.jpg Is it possible to determine this moment of inertia in this problem using double integrals of the form: http://img172.imageshack.us/img172/1219/momented0.jpg I could do this problem using double integrals if the...
  37. G

    Finding the Correct Function and Limits for a Polar Double Integral

    I'm having trouble finding the function and/or the limits to this problem: Using polar coordinates, evaluate the integral http://ada.math.uga.edu/webwork2_files/tmp/equations/01/19aeef09224e0fca11ef9d6e45fb311.png where R is the region...
  38. G

    Reverse Order Integration for Improper Double Integral

    Here's the question: We want to evaluate the improper integral http://ada.math.uga.edu/webwork2_files/tmp/equations/6c/4073055a5b909be16e2abc5bd3dfc61.png Do it by rewriting the numerator of the integrand as...
  39. K

    Solving a Proof without Double Integral: A Challenge

    hi how r u all i have a small problem with this proof i want the solution without using double integral that `s the proof http://s07.picshome.com/ce2/aaa.jpg
  40. I

    How do you solve the ff double integral?

    given \phi to be a function of x and t, how do you solve 2\int_{0}^{\infty}\int_{x}^{\infty}\frac{\partial^{2}\partial\phi}{\partial t^{2}} dt dx - 2\int_{0}^{\infty}\int_{0}^{t}\frac{\partial^{2}\partial\phi}{\partial x^{2}} dx dt any hints would be great. thanks!
  41. T

    Double integral (6x^2 -40y)dA

    double integral (6x^2 -40y)dA where it is a trianglewith vertices (0,3) , (1,1) and (5,3) may i know how to divide the region according to this triangle??
  42. S

    How do I correctly handle absolute value signs in double integral applications?

    Question: At airports, departure gates are often lined up in a terminal like points along a line. If you arrive at one gate and proceed to another gate for a connecting flight, what proportion of the length of the terminal will you have to walk, on average? One way to model this situation is...
  43. D

    Double Integral in Polar Coordinates problem

    I am having trouble with this seemingly easy problem. Evaluate the double integral (sin(x^2+y^2)) , where the region is 16=<x^2+y^2=<81. I find the region in polar coordinates to be 4=<r=<9 0=<theta=<2pi I find the expression to be sin(rcos^2theta+rsin^2theta) r dr dtheta , which is...
  44. U

    Double Integral Help: Solving with u-sub

    \int _0 ^{\pi/3} \int _0 ^{\pi/4} x cos(x+y) dy dx \int _0 ^{\pi/3} xsin(x+\frac{\pi}{4}) dx using u-sub, u=x, dv=sin(x+pi/4) -xcos\left(x+\frac{\pi}{4}\right)+ sin\left(x+\frac{\pi}{4}\right)-sin\left(\frac{\pi}{4}\left) |_0^{\pi/3}...
  45. B

    What are the limits for a double polar integral in the first quadrant?

    i have the integral \int_{0}^{\infty} \int_{0}^{\infty} (-x^2-y^2) \ dx dy (double integral with both limits the same...assuming my first bash at the tex comes out it says to transfer it into polar form and evaluate it i have no idea how to convert a limit of infinity to polar form, help...
  46. H

    Simplifying a Complicated Double Integral?

    Hi, I am new here, but apparently there are some decent mathematicians around, so I would like to confront you with a double integral problem. Consider \psi_n(z) = \int_0^{2\pi}\int_0^1 \frac{ (z-\frac{1}{2}) \cdot (r \cos(\theta) + \frac{1}{2})^n \cdot r} { \sqrt{...
  47. denian

    Double Integral Confusion: Why Can't I Use a Different Range for Integration?

    given f(x,y) = x-y R is a triangle with vertices (2,9), (2,1), (-2,1) then i need to find I, I = int. int. (x-y) dx dy i was taught to use this range when i do the dbl. integration 1 <= y <= 9 (y-5)/2 <= x <= 2 i want to ask, why can't i use this range: 1 <= y <= 9 -2...
  48. A

    How can I solve double integrals with tricky limits and substitutions?

    I was fine with these in class, tutorials etc. It's only since I found this in a past paper that I've had a problem with them. \[ \int_0^1\! \int_{\sqrt{y}}^1 9\sqrt{1-x^3}\,dxdy.\] Nomatter what I substitute in under the sqrt sign I just can't get out the integral for x :( I tried...
  49. R

    Integrating Over an Oval: Solving Double Integrals with Non-Circular Boundaries

    Anybody know how to integrate over... Z^2 = 4x^2 + y^2 with the plane z = 1 ? this comes from my class notes... hmmm.. the proff did some thing really messy... or at least i wrote it messy... but i got 0(integral)2pi 0(integral)1 z dz d(pheta) which doesn't seem to make...
  50. B

    Proving Change of Variables Formula for Double Integral w/ Chain Rule

    Hi, I'm having trouble understanding the solution to a question from my book. I think it's got something to do with the chain rule. The problem is to prove the change of variables formula for a double integral for the case f(x,y) = 1 using Green's theorem. \int\limits_{}^{} {\int\limits_R^{}...
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