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. C

    Surface Area of a Sphere without double integral

    Is it possible to come up with a derivation of the surface area of a sphere without using a double integral? Most of the ones I've found seem to involve double integrals; For example, this was given as the "simplest" explanation in a thread from 2005: S=\iint...
  2. R

    Calculate the double integral : int int xye^((x^2)(y))

    Homework Statement Calculate the double integral int int xye^((x^2)(y)) , 0<= x <= 1 , 0<= y <= 2 Homework Equations Integral by parts uv - int vdu The Attempt at a Solution The answer in the back of the book is (1/2)((e^2) -3) , but I get (1/2)((e^2) -1) . I think I made a...
  3. M

    Double integral volume of a region bounded by two z planes

    a) find the volume of the region enclosed by z = 1 - y^2 and z = y^2 -1 for x greater or equal to 0 and less than or equal to 2. b) would i split up the volume into two integrals, each integral for each z function and then add them together? I also don't know how to find the bounds...
  4. P

    Establishing double integral limits

    Homework Statement What would be the limits for each of the integrals (one with respect to x, one with respect to y) of an area bounded by y=0, y=x and x^2+y^2=1? Homework Equations None that I can fathom The Attempt at a Solution I've rearranged the latter most equation to get...
  5. W

    Double integral in polar coordinates

    Homework Statement I know I have the set up done correctly I am wondering where I went wrong because I know I cannot get zero, and I am a little worried I did my integration wrong. please help. http://i1341.photobucket.com/albums/o745/nebula-314/IMAG0107_zps3cde35a8.jpg
  6. E

    Can I change the limits of this double integral

    Homework Statement R is the region bounded by y=x^2 and y=4. evaluate the double integral of f(x,y)=6x^2+2y over R After drawing the region I was wondering if I could just work with the first quadrant and then double my solution, because both y=x^2 and y=4 are even functions so my question is...
  7. Z

    Calculate double integral of the intersection of the ellipse and circle

    how to calculate the double integral of f(x,y) within the intersected area? f(x,y)=a0+a1y+a2x+a3xy The area is the intersection of an ellipse and a circle. Any help will be appreciated, I don't know how to do this. can I use x=racosθ,y=rbsinθ to transformer the ellipse and...
  8. L

    Double Integral problem (E^x^3)

    Relevant equations The Attempt at a Solution i've tried changing the integration values from dxdy to dydx, but without success. i can't seem to get the same result after i change the ranges tried to change to 0<x<1 , x^2 < y <1 some light would be appreciated
  9. U

    Finding limits of double integral

    Homework Statement The problem is to solve the integral. First I did coordinate transformation by finding jacobian = (1/4)(x2 + y2). The problem is, I do not know the limits of integration after transformation...I tried using a graphical approach: by considering 2 cases: y>x and y<x and...
  10. DryRun

    Double Integral Theory - Exploring Concepts and Solutions

    Homework Statement There are 2 questions which deal with the concept of double integration. I think there's no need for any calculations, which might have been easier, in my opinion. 1. 2. http://img2.uploadhouse.com/fileuploads/17065/170654043ae9d827241bff097ca2ee9760242ef0.png Homework...
  11. DryRun

    Find double integral over region R

    Homework Statement Homework Equations For example, for f(x,y)=x+y-2 The Attempt at a Solution I've figured out part (a) which is quite simple. I simply used the relevant equations above for ##f(x,y)= 3(x^2+y^2)## I know i should use the given hint to figure out the value of I, which is...
  12. P

    Finding Volume Using Double Integrals: A Question on Cylindrical Coordinates

    Hi, Homework Statement I am asked to find the volume under the curve whose equation is z=16-(x^4+y^4), and within (x^2+y^2)<=1, using a double integral. Homework Equations The Attempt at a Solution Should I use cylindrical coordinates? I feel slightly lost. I have tried drawing...
  13. N

    Limits after mapping in double integral

    Homework Statement I have the double integral, ∫∫sqrt(x^2+y^2) dxdy, and the area D:((x,y);(x^2+y^2)≤ x) Homework Equations The Attempt at a Solution By completing the squares in D we get that D is a circle with origo at (1/2,0), and radius 1/2. Then I tried changing...
  14. N

    Double integral limits after mapping

    I have the double integral, ∫∫sqrt(x^2+y^2) dxdy, and the area D:((x,y);(x^2+y^2)≤ x) By completing the squares in D we get that D is a circle with origo at (1/2,0), and radius 1/2. Then I tried changing the variables to x=r cosθ+1/2, y=r sinθ and J(r,θ)=r which leads to a not so nice...
  15. S

    Can Trig Identities Help with Evaluating a Double Integral?

    Evaluate the integral: \int_0^\pi \int_x^\pi \frac{sin(y)}{y} Look, I've been at this problem for near an hour and a half. I've tried by parts, but I just get stuck in a loop. And I can't think of any way to do this. I've been reading things about taylor expanding it in order to...
  16. L

    Solving a Double Integral: 0.23 Numerically, Analytically Possible?

    Homework Statement Solve double integral \int^1_0\int^1_x\sin(y^2)dydx Homework Equations The Attempt at a Solution I got with Wolfram Mathematica 7.0 result 0.23 numerically. Can it be solved analyticaly?
  17. E

    Problem with limits of integration - converting double integral to polar form

    Homework Statement \int_0^2 \int_0^\sqrt{2x-x^2} xy,dy,dx I know the answer, but how does the 2 in the outer integral become pi/2?? I'm fine with everything else, I just can't get this...
  18. F

    Is the double integral convergent?

    Evaluate the integral ∫(2,∞) ∫(2/x,∞) 1/(y^2)*e^(-x/y) dydx by changing the order of integration. I get ∫(1,∞) ∫(2y,∞) 1/(y^2)*e^(-x/y)dxdy etc. etc. etc. I get to ∫(1,∞) (e^(-2)/y) dy Which is (ln∞-ln1)/e^2 = ∞ Does this thing not converge?
  19. STEMucator

    Quick double integral question

    Homework Statement Find the volume of the region R between the surfaces z = 4x^2 + 2y^2 \space and \space z = 3 + x^2 - y^2 Homework Equations The Attempt at a Solution Okay so I think I have an idea about how to do this one. First I check when the two surfaces intersect, that is when 4x^2 +...
  20. STEMucator

    Extremely difficult double integral.

    Homework Statement \int_{0}^{8} \int_{y^{1/3}}^{2} \frac{1}{x^4+1} dxdy Homework Equations Completing the square. The Attempt at a Solution This integral is disgusting. It literally took me 4 sheets of paper to do the partial fraction decomposition and then integrate the inner...
  21. A

    Stokes Theorem;determine double integral

    Homework Statement Let S be the surface defined by y=10 -x^2 -z^2 with y≥1, oriented with rightward-pointing normal. Let F=(2xyz+5z)i+ e^x Cos(yz) j +x^2 y k Determine ∫∫s ∇×F dS (Hint: you will need an indirect approach) Homework Equations Stokes Theorem ∫∫s ∇×F dS The...
  22. B

    Double Integral and Polar, Really Need Help in the next few hours

    I have this problem and I cannot even begin to start it. I have to hand it in today in a few hours, and I have been stuck on it for what seems like for ever. It reads: By using polar coordinates evaluate: ∫ ∫ (2+(x^2)+(y^2))dxdy R where R={x,y}:(x^2)+(y^2)≤4,x≥0,y≥0} Hint: The...
  23. M

    Double Integral over general region

    Homework Statement Find the integral using a geometric argument. ∫∫D√(16 - x2 - y2)dA over the region D where D = {(x,y) : x2 + y2 ≤ 16} By the way, the subscript D next to the integral refers to the region over which the function is integrated.Homework Equations ∫∫f(x,y)dxdy = ∫∫f(x,y)dydx...
  24. C

    Double integral and polar cordinates other problem.

    If we have to find the volume, written in polar cordinates, inside this sphere X2+y2+z2=16 and outside this cylinder x2+y2=4 How should I approach this? Could I take the sphere function and reqrite in polar cordinates z=√(16-X2-y2) which is the same as z=√(16-r2) But then I have...
  25. C

    Double integral polar cordiantes

    Hi, I need help with this problem Evaluate the given integral by changing to polar cordinates ∫∫xydA where D is the disc with centre the origin and radius. My solution so far. I believe this would give a circle with radius 3 in xy plane. And then x=r*cos(θ) and y=r*sin(θ) So...
  26. A

    Calculating Volume Between Paraboloids and Cylinder

    Find the exact volume of the solid between the paraboloids z=2x ^{2}+y ^{2} and z=8-x ^{2}-2y ^{2} and inside the cylinder x ^{2}+y ^{2}=1. I really don't know how to set this up. Would it be something like ∫∫(2x^2+y^2)-(8-x^2-2y^2)dA + ∫∫(x^2+y^2-1)dA ? If so, how would I find the bounds...
  27. W

    Volume using double integral (polar coordiantes)

    Homework Statement use a double integral to find the volume of the solid bounded by. z=x^2+2y^2 and z=12-2x^2-y^2 I want to change variables using polar coordinates, I know its the top minus the bottom, and the intersection between the two is a circle radius 2. The Attempt at a...
  28. M

    Double integral using polar coordinates

    The question is in the paint document I wanted to know why they integrated from 0 to pi and not from 0 to 2pi
  29. W

    Double Integral with Trigonometric Functions: Troubleshooting and Evaluation

    Homework Statement \int^{\pi}_{0} \int^{1-sin\theta}_{0} r^{2} cos\theta drd\theta I keep getting an answer of 0 but i am most certain that i am getting my trig messed up somewhere. 1/3 \int^{\pi}_{0} r^{3} cos\thetad\theta from 0 to 1-sin\theta then i get 1/3...
  30. M

    Conceptual double integral question

    Homework Statement ∫∫x2sin(y2)dA; R is the region that is bounded by y=x3 y=-x3, and y=8. While working out the regions for this integral I set the inner integral to -y1/3 to y1/3. and the outer integral from 0 to 8. The book however set the inner integral from 0 to y1/3. However we both...
  31. M

    Double Integral Homework R1 & R2

    Homework Statement Homework Equations The Attempt at a Solution The question is in the picture, the new integral I got was R1:0≤y≤x R2:0≤x≤1 However the answer is R1: is inner integral R2: is outer integral I drew a graph showing my thought process...
  32. D

    MHB Double Integral A₁, -1 - Does it Check Out?

    \begin{alignat*}{3} A_{1,-1} & = & \frac{50\sqrt{3}}{\sqrt{2\pi}}\int_{-\pi/4}^{\pi/4}\int_0^{\pi}e^{-i\varphi}\sin\theta d\theta d\varphi\\ & = & \frac{100\sqrt{3}}{\sqrt{2\pi}}\int_{-\pi/4}^{\pi/4}e^{-i\varphi}d\varphi\\ & = & \frac{100\sqrt{3}}{\sqrt{\pi}}\\ \end{alignat*} Is this correct?
  33. H

    Cannot finish calculating a double integral with change of coordinates

    Homework Statement Integrate: \displaystyle f\left( x,y \right)=\frac{{{x}^{2}}}{{{x}^{2}}+{{y}^{2}}} on the region: \displaystyle D=\left\{ \left( x,y \right)\in {{\mathbb{R}}^{2}}:0\le x\le 1,{{x}^{2}}\le y\le 2-{{x}^{2}} \right\} TIP: Use change of coordinates: \displaystyle...
  34. J

    Is there a simple rule for double and multiple integration?

    Hi, I have a very limited knowledge of calculus, and even less of integration. However I know that the general rule for integration is x^{n} integrates as \frac{x^{n+1}}{n+1} Is there a similar rule for double integration and is there a rule that can be extended to...
  35. M

    How to Set Up Limits for a Double Integral Over a Complex Region?

    Homework Statement ∫∫x2dA; R is the region in the first quadrant enclosed by xy=1, y=x, and y=2x. First thing I did was notice that I had to find dydx, then I graphed y=1/x, y=x, and y=2x. Graphing I say that the limit of dy lie between x≤y≤2x However I get confused as to how...
  36. H

    What is the Best Approach for Solving a Complicated Double Integral?

    Homework Statement I've got to calculate: \displaystyle\int_0^1\displaystyle\int_0^x \sqrt{4x^2-y^2} dy dx Homework Equations The Attempt at a Solution I've tried the change of variable: \displaystyle t=4{{x}^{2}}-{{y}^{2}} but it doesn't get better. I've also tried polar...
  37. M

    MHB Calculate area with double integral.

    Hello all, I haven't been on here for a while. I'm glad to see that everything is picking up nicely. Anyway, I have a question that I see the answer to, but I am not understanding the concept. Find the area of the region bounded by all leaves of the rose \(r=2\cos(3\theta)\) The thing I am...
  38. J

    How to Add Upper and Lower Limits to Integrals in Forum Math Code?

    How does this work? Like, is it integrating the integral of f(x)? Kind of like... a higher order integral? I've seen these problems before, kind of confusing; Lol random thought: InteCeption.(Also, how do I add upper and lower limits to integrals with your forum math code thing?) \int \int...
  39. S

    Double integral over a circular region using rectangular coord's

    I would like to compute $$ \iint \limits_{x^2 + y^2 \le 3} \! x^2 + y^2 \, \mathrm{d} A $$ using rectangular coord's. First, I'll compute the iterated integral using polar coordinates so that I can check my work. Limits: $$ 0 \le \theta \le 2\pi \\ 0 \le r \le 3 $$ so $$ \iint \limits_{x^2 +...
  40. D

    Definite double integral of e^x^2

    Hi guise. I just encountered a problem which I sincerely don't know how to attack. I don't know what kind of variable substitution would help me to solve this problem... It's that goddamn e^x^2 which is a part of the integrand... I don't know if I should use polar coordinates either... Please...
  41. N

    Double integral, change of variables or no

    Homework Statement ∫∫Se2x+3ydydx where S is the region |2x|+|3y|≤ 1 Homework Equations The Attempt at a Solution So I've done this two ways and gotten two different answers and I'm not sure which is right. I used change of variables where where u=3y+2x and v=3y-2x and I got an...
  42. G

    What is the Double Integral Problem for the given function?

    Homework Statement \int_{0}^{1}\int_{0}^{1} xy \sqrt{x^2 + y^2} dy dx Homework Equations The Attempt at a Solution So I tried integration by parts, but I'm not really coming up with anything simpler. I also thought I could use a u substition, letting u= x^2+y^2, but then it was...
  43. R

    Double integral to find area of a portion.

    Homework Statement the question is 3(b) on the attached pdf. Homework Equations The Attempt at a Solution I could only get as far as the filling in the equation. How do they change it to one integral.? And also where did they get them substitutions from? Any help...
  44. C

    Proving the Double Integral Problem: Reversing the Order of Integration

    The question is: Show that: \int_0^1\int_x^1e^\frac{x}{y}dydx=\frac{1}{2}(e-1) I've tried reversing the order of integration then solving from there: \int_0^1\int_y^1 e^{\frac{x}{y}}dxdy =\int_0^1[ye^\frac{x}{y}]_y^1dy =\int_0^1ye^\frac{1}{y}-ye^1dy But I can't integrate...
  45. E

    Help with double integral problems

    Can someone explain to me how I would arrive at this answer?: http://www.wolframalpha.com/input/?i=double+integral+of+cos%28x%5E2%29dxdy+from+x%3Dy+to+x%3Dsqrt%28pi%2F2%29+and+y%3D0+to+y%3Dsqrt%28pi%2F2%29 This double integral problem came up in a practice test I was taking, and I just can't...
  46. S

    U sub of a real double integral

    Homework Statement ∫∫ 1 / (2x + 3y), R = [0,1] x [1,2] Homework Equations - Iterated integrals - u sub The Attempt at a Solution Here is my attempt at solving this (I must be screwing up on the algebra) Integrating with respect to x u = 2x, dy = 2 u^-2 du u^-2 =...
  47. Avatrin

    Why is this a type II double integral?

    ∫∫_{A}xy^{2}dxdy A is the area between y = x^2, y = 2-x and x\geq0. I am told that this is a type II double integral and I thus have to: ∫^{1}_{0}∫^{2-y}_{√y}xy^{2}dxdy But, why can't I do this? ∫^{2}_{0}∫^{2-x}_{x^2}xy^{2}dydx
  48. R

    Weird double integral. Please help

    Weird double integral. Please help! its from thermodynamics...but i don't think you really need to understand thermodynamics to figure out what math trick they used to get from the first integral to the second integral http://img833.imageshack.us/img833/833/intek.png i have been looking...
  49. D

    Reverse order of double integral

    Homework Statement Integrate (x+2y) over y=1+x^2 , y=2x^2 and x=0, x=1 (dy dx) Homework Equations Graph is sketched. The Attempt at a Solution y = 2x^2 --> x=(y/2)^(1/2) y = 1+x^2 --> x=(y-1)^(1/2) integrate over y=0 to y=2 problem encountered when solving definite integral from y=0 to...
  50. S

    Double Integral with Limits: Solving for (x+2y)^-(1/2)

    Homework Statement Calculate the integral: ∫ ∫_R (x+2y)^{-(1/2)} dxdy Where R is defined as points (x,y) which satisfy: x-2y ≤ 1 and x ≥ y2 + 1 Homework Equations So basically I'm completely stuck on this exerzice. As far as I can see, you could make the x limit go from...
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