Double integral Definition and 573 Threads

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. R

    Double Integral of min(x,y): How Do I Solve This Homework Problem?

    Homework Statement \int\intmin(x,y)*dx*dy both limits are from 0 to 1 what is the general solution of this question ? Homework Equations The Attempt at a Solution I attempted to solve by saying that x<y, finding the integral and then, y<x and finding its integral. meaning...
  2. S

    Correcting Limits in Double Integral for Intersection Volume of Three Cylinders

    Homework Statement I was trying to find the volume of the intersection between 3 cylinders x^2 + y^2 = 1, y^2 + z^2 =1, and z^2 + x^2 =1. I set up the double integral in two different ways: 8\int_{\theta = \frac{-\pi}{4}}^{\frac{\pi}{4}}\int_{r = 0}^{1} \sqrt{1 - r^{2}\: cos^{2}\, \theta}\...
  3. M

    Double Integral Plus Integration by Parts with Natural Log Problem

    Homework Statement My homework problem is the double integral of y/1+xy dxdy. It is a definite double integral and both integrands have the values of a = 0 and b = 1. Homework Equations Integration by parts: uv - int(vdu) The Attempt at a Solution My first step of the double integral is I...
  4. J

    What is the double integral for a triangle with vertices (1,1),(2,4),(5,2)?

    Homework Statement Compute the double integral where the region is a triangle with vertices (1,1),(2,4),(5,2).(please see the attachment) Homework Equations double integral((16-2x-3y)1/11)da The Attempt at a Solution first i found the equation of the three lines ...
  5. S

    Double Integral: Tricky Exponential Integration

    Homework Statement Calculate the following: integrate [ (e^(x/y)) dy y=x^(1/3)..x ] dx x=1..8Homework Equations The Attempt at a Solution In this current state the dy part is not integrable, so there must be some trick with changing the variables or something else. Couldn't find a proper...
  6. V

    Where Should the Weather Station Be Located in the Preserve?

    Homework Statement The center of a national park is located at (0,0). A special nature preserve is bounded by by straight lines connecting the points A at (3,2), B at (5,1), C at (8,4) and D at (6,5) in a parallelogram. The yearly rainfall at each point is given by RF(x,y)=x^2+xy+y^2 in...
  7. B

    Reversing Order of Integration: Double Integral Evaluation

    Evaluate the integral by reversing the order of integration. I'm not exactly sure how to approach this problem. Any help would be appreciated!
  8. G

    Proving a Double Integral Identity: Can Anyone Help with This Homework Problem?

    Homework Statement Can anybody prove the following double integral identity? How?: \int_{0}^{1} s(1-s) f(sx) ds = \int_{0}^{1} s^2 \int_{0}^{1} t f(tsx) dt ds Here f(x) is an arbitrary Riemann-integrable function. Thanks in advance. Homework Equations I've found the following but it...
  9. S

    What is the Correct Domain of Integration for a Double Integral Problem?

    Homework Statement question 2: http://www.math.ubc.ca/~haber/courses/math253/Welcome_files/asgn5.pdf" The Attempt at a Solution So for part a) I tried to plot my domain of integration and ended up concluding it was an area bounded by y=0, y=4-x^2, and x=1. Is this okay? In not too...
  10. Telemachus

    How to properly solve a double integral for calculating volume?

    Homework Statement Hi there. I have this problem with double integral, which says: calculate using a double integral the volume limited by the given surfaces: x^2+y^2=4, z=4-y z=0 Its a cylinder cut by a plane. At first I've did this integral...
  11. K

    Unraveling a Complicated Double Integral

    URGENT - Help with a double integral! Homework Statement This is a statistics problem for another question but it involves a somewhat complicated double integral. So I am stuck at the last line. We were told to use the substitution with u as i wrote down. I don't know what to do...
  12. Telemachus

    Double Integral Doubts: Understanding Regions and Order

    Homework Statement Hi there, I've got this doubt about a double integral. I have this region: \displaystyle\int_{-1}^{2}\displaystyle\int_{-\sqrt[ ]{4-x^2}}^{1-x^2}f(x,y)dydx And the thing is, how this region would look like? Would it look like this?: The thing is that after the cut...
  13. N

    How Can You Evaluate This Complex Double Integral?

    \int_{1}^{2}\int_{y}^{y^3}e^{\frac{x}{y}}dxdy \int_{1}^{2}ye^{y^2}-eydy taking u = y^2 \frac{1}{2}\int_{1}^{4}e^udu-\int_{1}^{2}eydy e^4/2-2e
  14. C

    Help with understand double integral solution?

    http://img844.imageshack.us/img844/3293/17693169.jpg I follow it up to the third step, but how did they get the bottom denominators of -4? shouldn't it be -1?
  15. N

    Evaluate Double integral over triangular region

    Homework Statement Evaluate the double integral. [I don't know how to write it in latex, sorry, but its the double integral over the region 'd' of ye^x dA D is the triangular region with vertices (0,0), (2,4), and (6,0). Homework Equations The Attempt at a Solution So the...
  16. B

    Show that double integral does not exist

    Homework Statement Here are my problems.[PLAIN]http://img256.imageshack.us/img256/2254/whatua.jpg Homework Equations possible conversions to polar coordinates but I doubt that's needed. Fubini's theorem?The Attempt at a Solution So for the h(x,y) integral does not exist, I got this: i...
  17. B

    Double integral help please? polar and cartesian

    Homework Statement Okay here's the problem: Consider the region R interior to a circle(of r =2) and exterior to a circle(r=1). 1.Using cartesian coords and double integral, calc the area of annulus. 2. repeat calculation above but using double integral with polar coords The...
  18. R

    Double integral in polar coordination

    Homework Statement Homework Equations The Attempt at a Solution Please tell me if I am wrong. I suspect about the ranges. Are my range corrrect?
  19. J

    Double Integral Volume Problem

    Homework Statement Use double integrals to find the volume of the region in the first octant (x, y, z all more than or equal to zero) bounded by the vertical plane 2x + y = 2 and the surface z = x2 Homework Equations The Attempt at a Solution I'm having major problems visualizing...
  20. jegues

    Understanding Double Integrals: Finding the Area Between Two Circles

    Homework Statement See Figure. Homework Equations N/A The Attempt at a Solution Simplifying the double integral, \int \int_{R} \sqrt{1 + 4x^2 + 4y^2} dA Am I suppose to put in the bounds for part a, as part of simplifying the integral? This brings me to part b along with...
  21. J

    Double integral of product of Diracs

    Hi there! I am having a bit of a trouble when I try to work out a demonstration involving Dirac delta functions. I know, they are not real functions, and all that, but it only makes my life more difficult :) Lets begin by the beginning to see if anyone can help. The first equation I will...
  22. E

    Double Integral with a trig function

    just wondering if i can still do this, attempted the following: ʃʃ cos(x+y)dxdy with upper limits of pi/2 and lower limits of 0 for both integrals My answer came out as 0. Can anyone confirm this?
  23. M

    Double Integral Confusion: How Do I Handle Boundaries with U-Substitution?

    i am confused about the double integral ʃʃ cos(x+2y)dA, where R = [0,pi]x[0,pi/2] i realize for the integral that i must do u-substitution. when i do this, however, do i also have to change the boundary conditions as in a single integral? i got -8 without changing the boundary conditions...
  24. agro

    Reexpressing double integral using parametric functions

    If: x = f(t) (continuous and differentiable) y = g(t) (continuous) x is nondecreasing on [a, b] y is nonnegative on [a, b] Then when we trace the points (x,y) from t=a to t=b, we can calculate the area bounded above by the traced curve (below by y = 0, left by x = f(a), and right by x = f(b))...
  25. agro

    Double integral in polar coordinate

    Homework Statement With a > 0, b > 0, and D the area defined by D: \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 Change the integral expression below: \iint\limits_D (x^2+y^2) dx\,dy by using x = a r cos θ, y = b r sin θ. After that evaluate the integral. The Attempt at a Solution...
  26. A

    Double integral, change order of integration, int(0,1)int(y,1)(e^(-x^2)*y^2)dxdy

    hey, i'm having some difficulties solving a problem. i want to know exactly how to go about solving it, since i am studying for a final exam. i know that i need to change the order of integration, but i'd also like to see how it's done correctly, since no official answers are provided... (my...
  27. rubenvb

    Double integral to single by magic substitution

    double integral to single by "magic" substitution Hi, I have a double (actually quadruple, but the other dimensions don't matter here) integral which looks like this: \iint_0^\infty \frac{d^2 k}{k^2} Now, someone here told me to replace that with \int_0^\infty \frac{1}{2} 2\pi...
  28. A

    Setting up a double integral to find the volume

    Homework Statement Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equation x2+y2+z2=r2 Homework Equations Not much equations, just setting the integral up, however I have no idea. The Attempt at a Solution I know...
  29. 8

    Integrating a Quarter Circle with a Double Integral

    Homework Statement I=\int^{a}_{0}dx\int^{\sqrt{a^{2}-x^{2}}_{0}(x-y)dy Homework Equations r^{2}=x^{2}+y^{2} The Attempt at a Solution Im thinking that the question is asking to integrate the first quarter of the circle of radius a between 0 and pi/2. In that case I've changed...
  30. E

    Double Integral - Going from Cartesian to Polar

    Homework Statement See attachment. Change the Cartesian integral into an equivalent polar integral, then evaluate the integral. I have no problems at all converting the actual function I am integrating or the integration itself, it is just the limits I cannot do. I've posted two...
  31. S

    Finding Volume Under Cone Above Disk

    Homework Statement Use polar coordinates to find the volume of the given solid: Under the cone z = Sqrt[x^2 + y^2] Above the disk x^2 + y^2 <= 4 2. The attempt at a solution I tried using formatting but I couldn't get it right so I'll explain...I changed variables by making the upper and...
  32. S

    Double integral over circle region

    Homework Statement So I have to use the type I type II region formula to find the volume under the equation (2x-y) and over the circular domain with center (0,0) and radius 2. Do I have to split this circle into semi-circles and treat it as 2 type I domains? I got the following limits for...
  33. S

    How do I find the volume under a circular domain using double integrals?

    So I have to use the type I type II region formula to find the volume under the equation (2x-y) and over the circular domain with center (0,0) and radius 2. Do I have to split this circle into hemispheres and treat it as 2 type I domains? I got the following limits for the top half, but I get...
  34. U

    Evaluating a Reversed Order Double Integral

    Homework Statement Eveluate by reversing order of integration \int^{2}_{0}\int^{1}_{y/2} ye^{x^3}dxdy Homework Equations The Attempt at a Solution this is what I got... \int^{1}_{0}\int^{2x}_{0} ye^{x^3}dydx I end up with... \int^{1}_{0} 2x^2e^{x^3}dx I don't know...
  35. M

    Solving a Difficult Double Integral: Tips and Tricks for Success

    Homework Statement \int_{0}^{1} \int_{0}^{1} \sqrt{4x^2 + 4y^2 + 1} dx\,dy The Attempt at a Solution This integral is tough for me, I couldn't think of a way to evaluate it. Can you suggest me the first step to do this problem? Thanks!
  36. P

    Evaluate the following double integral

    Homework Statement Change the order of integration and evaluate the following double integral: I = {\int_0^{1} \left({\int\limits_{y}^{1} 30 y\sqrt{1+x^3} \mathrm{d}x }\right) {\mathrm{d}y} So thenn i did = 30 \int_0^{1} \sqrt{1+x^3} \left({\int_0^{x} y \mathrm{d}y}\right)...
  37. S

    Double Integral of yx: Solving with k^2*X^2*a^3/6 - Attempt at Solution

    Homework Statement http://img687.imageshack.us/img687/6092/dvojniinteg3.png Uploaded with ImageShack.us The Attempt at a Solution =k^2*X^2*a^3/6 Is this the correct solution?
  38. B

    Evaluating double integral - jacobian help

    Homework Statement trying to evaluate the double integral from 0 to infinity and 0 to infinity of [(x^2 + y^2)/1 + (x^2-y^2)^2]e^-2xy dxdy using the coordinate transformation u=x^2-y^2 and v=2xy Homework Equations The Attempt at a Solution so i calculated the jacobian...
  39. R

    Changing the Order of Integration for a Double Integral: How Do I Evaluate This?

    Homework Statement Looks like I'm back with another question already :frown: I need to change the order of integration for this double integral and then evaluate it, but I get to a point where I'm not sure what to do. Homework Equations \int^3_{0} \int^9_{y} \sqrt{x}cos(x) dx dy The...
  40. T

    Solve Tricky Double Integral for Charge Density Distribution in Symmetric Disc

    Homework Statement I'm trying to model the potential field in and around a symmetrically charged disc where the charge density drops exponentially from the center. Homework Equations This can be done by solving the double integral: \int ^{2 \pi} _{0} \int ^{\infty} _{0} \frac{r e^{-r/b}...
  41. F

    Volume using double integral and polar coordinates

    Find the volume under the cone z = sqrt ( x2+y2 ) and on the disk x2+y2 < 4. Use polar coordinates. Graphing x2+y2 < 4, I get a circle centered at 0,0 with radius of 2 So theta goes from 0 to 2pi Also, since x2+y2 < 4 This means that r^2 < 4 so -2 < r < 2...
  42. I

    What are the limits of integration for evaluating a double integral?

    Homework Statement Evaluate the following double integral: ∫ ∫ R sin (x/y) dA where R is the region bounded by the y axis, y=pi and x=y^2 Homework Equations as in problem statement The Attempt at a Solution Well I started this question by drawing the area to be evaluated...
  43. E

    Double Integral - polar coordinates

    Homework Statement \displaystyle\int\int\sqrt{4-x^2-y^2} dA R{(x,y)|x^2+y^2\leq4 .. 0\leq x} The Attempt at a Solution So far i have: \displaystyle\int^{\pi}_{0}\int^{r}_{0}\sqrt{4-r^2} rdrd\theta Solving i get...
  44. R

    Convert triangle vertices to double integral polar coordiantes

    Homework Statement integrate f(x,y) = sqrt(x^2+y^2) over triangle with vertices (0,0) (0,sqrt2) (sqrt 2, sqrt 2) Homework Equations x= rcosO, y = rsinO x^2+y^2=r^2 The Attempt at a Solution im supposed to use a double integral converted to polar coordinates, so i used...
  45. S

    Double integral in polar coordinates

    Homework Statement Ok so I solved the problem, I think. I would just like to check my work. So the problem is: Use polar coordinates to find the volume of the given solid bounded by the paraboloids z = 3x^2 + 3y^2 and z = 4 - x^2 - y^2. Homework Equations r^2 = x^2 + y^2 x = r cos...
  46. E

    Double integral - reversing order

    Homework Statement \displaystyle\int^1_0 \int^{e^x}_{1}dydx Homework Equations noneThe Attempt at a Solution the above integral i can do with no problem, but changing the order of integration give me a totally different answer and need to know if i am doing it correct...
  47. K

    Double Integral Substitution Techniques for Evaluating Complex Integrals

    Homework Statement Evaluate the integral. 1|0 s|0 ( t . sqrt ( t2 + s2 ) dt dsI hope the way I've written it makes some sort of sense. The Attempt at a Solution After getting my head around changing the order of integration I get hit with this question and for some reason am totally...
  48. J

    Volume in a cone, using a double integral.

    Homework Statement Evaluate the volume under z^2 = x^2 + y^2 and the disc x^2 + y^2 < 4. Just wondering what I should write to constitute a proper solution. Would this do?: V=(int)(int) z dA R is {x²+y² < 4} [context: R in other problems was the region over which integrals were performed]...
  49. L

    Evaluating Double Integral Involving Imaginary Error Function

    Homework Statement Evaluate \int\int x^{2}e^{x^{2}y} dx dy over the area bounded by y=x^{-1}, y=x^{-2}, x=ln 4 Homework Equations The Attempt at a Solution \int^{1}_{(ln 4)^{-2}}\int^{y^{-1}}_{y^{\frac{-1}{2}}}x^{2}e^{x^{2}y}dx dy I got this far before I realized that this wasn't a...
  50. B

    Reversing order of integration of double integral qns.

    [sloved]reversing order of integration of double integral qns. Homework Statement pls refer to attached picture. Homework Equations The Attempt at a Solution intially upper and lower limits are , x^2 < y< x^3 and -1<x<1 sketched y=x^2 and y= x^3. => sqrt(y) =x and cube root...
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