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. M

    Double integral with polar coordinates

    Homework Statement It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ. f(x,y)=1 (plane parallel to Oxy plane) They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it. Homework Equations x=rcosθ y=rsenθ r=√x2+y2 The Attempt at a...
  2. M

    Mathematica Mathematica: draw a double integral

    I've written a MATLAB m-file to draw a double integral as below.My problem is how to draw a double integral when the variables are located in the bounds of the integral not under its operator! m-file: tetha = pi/4; lamb = -1; h = 4; tetha0 = 0; syms x y l n =...
  3. B

    Double Integral with Strange Limits of Integration

    Homework Statement For the double integral ∫[0,1]∫[0,x^3] e^(y/x) dxdy (a) sketch the region of integration (b) evaluate the integral and (c) re-express the integral with the order of integration reversedHomework Equations NoneThe Attempt at a Solution The problem is that I've never seen a...
  4. J

    Double Integral Cartesian to Polar Coordinates

    Homework Statement Use polar coordinates to evaluate: ∫sqrt(2)0 ∫sqrt(4-y2)y 1/(1+x2+y2) dxdy Homework Equations The Attempt at a Solution I graphed it and I see r is the part of the elipse sqrt(4-y2) and goes from 0 to ∏/4. I'm not sure how to make the bounds for r or how to...
  5. T

    Double Integral: Calculating Limits & Value

    Homework Statement I have to calculate the double integral given below. Homework Equations ∬D√(x2 +y2) dxdy,D=x2+y2≤2x The Attempt at a Solution How can i calculate the limits from the give inequality to calculate the value of the given double integral? Waiting for a kind...
  6. C

    Double integral with substitution

    Homework Statement Evaluate (using a substitution) \int\int_{B}x^{2}+2y dxdy where B=\{(x, y) | x^{2}+y^{2}≤4\} The Attempt at a Solution I attempted a solution using polar coordinates, so the integral becomes \int\int_{B_{ρθ}}(ρ^{2}cos^{2}(θ)+2ρsin(θ)) ρ dρdθ, and the integration...
  7. C

    Double integral on one function of x and another of y

    Hi there. I think I have proved on little theorem on double integrals, showed below. Are my arguments 'correct' (I mean, rigorous enough)? Let f be a function of x, f(x), g be a function depending only on y, g(y), and last, let A be the set determined by a≤x≤b and c≤y≤d. By Fubini's theorem...
  8. N

    How to Correctly Calculate This Basic Double Integral?

    Homework Statement Integrate the following function f over the given region Ω: f(x,y) = xy; Ω bounded by y = 0, x = 2a, and x^2 = 4a The Attempt at a Solution The given answer is (a^4)/3 The answer I got was (a^4)...ignore the answer I put in the attachment at the end, slight...
  9. K

    Set Up Double Integral to find Vol. Solid, Bounded by Graphs

    Homework Statement ...Bounded by graphs of equations: z=xy, z=0, y=x, x=1 I don't know what z=xy is. The rest of boundaries are clear. I assume that when y=1 and x=1, z=1. But, is this a z=1 plane? Check my figure attached. Thank you. Homework Equations The Attempt at a Solution
  10. B

    Formula involving double integral over a disc?

    Problem: Can anyone help me out with the following problem: I am given a uniformly continuous function : g:\mathbb{R}^{2}\rightarrow [0,\infty ) such that the following condition is satisfied: \sup_{r> 0}\iint_{{x^{2}+y^{2}\leq r^{2}}}g(x,y)dxdy< \infty The question is to prove that:\lim_{|...
  11. C

    Double integral over the area of a square

    Homework Statement Find the area of a square with each side measuring 1 using double integral and change of euclidean coordinates to polar coordinate. Homework Equations x=rcosθ y=rsin0 dA=dxdy=rdrdθ The Attempt at a Solution int(int(rdr)dθ)
  12. H

    Numerical computation of double integral

    Hi, I wrote a piece of MATLAB code to compute a double integral of the form: \int_{a}^{b}\int_{c}^{d}f(x,y)dxdy I went about it using the trapezium rule, so what I did was apply the rule to the x variable first to obtain: \int_{a}^{b}\int_{c}^{d}f(x,y)dxdy\approx\frac{\delta...
  13. H

    How Do I Integrate cos(x/y) with Respect to y in a Double Integral?

    Suppose we have: ∫∫ cos(x/y) dydx where the first integral is of x and is 0→1, while the second is of y and is x→1. Could someone tell me how to get the first integration (i.e. of cos(x/y) w.r.t. y) done??
  14. H

    What is the Correct Answer for the Double Integral of exp(y)/y?

    Suppose the question is: ∫∫exp(y)/y dydx Now the first integral is w.r.t. y and goes from 0 to 1. The second integral is that of x and goes form y to y^2. I've evaluated it multiple times and it comes out to be 2-e. ( I first integrate the integrand w.r.t. x then w.r.t. y as the other...
  15. A

    Evaluating the Double Integral of F(bar) in C

    \int xy^3 dx+ x^5 dy, where C is the rectangle with vertices (0,0), (4,0), (4,2), and (0,2) F(bar)= <P,Q> <xy^3, x^5> derivative of P with respect to y= 3xy^2 derivative of Q with respect to x= 5x^4 Double \int (5x^4-3xy^2) dx dy with limits for x from (0,2) and y limits (0,4) I get 0 for...
  16. P

    Double integral over region surrounded by two ellipses

    Homework Statement A thin plate has the form of the intersection of the regions limited by \frac{x^2}{9} + \frac{y^2}{4} = 1 and \frac{x^2}{4} + \frac{y^2}{9} = 1 Which is the plate's mass if his density is δ(x, y) = |x| 2. The attempt at a solution I've tried using u, v...
  17. H

    Different uses of the double integral?

    Could somone tell me how is it that the double integral could be used for both calculating the area as well as the volume? And please explain that how does the triple integral, which is used to find the volume as well, fits in the picture and how is it different from the double integral we use...
  18. P

    Double integral finding the area

    Homework Statement Use a double integral to find the area of the region bounded by the curve r= 1+sin(theta)? Homework Equations The Attempt at a Solution I can't figure out what theta is intregrated from. I've tried from -(pi)/2 -> +(pi)/2 and that doesn't work. I've also tried...
  19. K

    Can TdS be computed using an area, time double integral of heat flux density?

    Can the TdS by a system be calculated by taking the area, time double integral of the heat flux density? If so, is it possible that this double-integral would take on an opposite sign if inside there was a dominating, growing black hole, where, I would presume, heat flows inwards, and not...
  20. DryRun

    Evaluating Double Integral for Region Ω

    Homework Statement Evaluate \int\int (2x+1)(x-y)dxdy where Ω is the region in the first quadrant of the x-y plane bounded by: y=x, y=x+2, y=2-x^2 and y=4-x^2 The attempt at a solution This is the graph of the intersecting curves and the shaded area is the region Ω...
  21. D

    Limits for double integral over trapezoidal shape

    Homework Statement I'm trying to determine the limits for a double integral over a symmetric trapezoid or equilateral triangle. I'm not trying to determine the area, and therefore using symmetry to simplify the integration is not an option. The limits for the integration over the y-axis are...
  22. V

    How to Simplify a Double Integral with Exponential and Power Functions?

    Homework Statement $$\left[\int\limits_0^{Inf} {\int\limits_0^{Inf} {e^{ - aX - bY} \cdot F(X + Y + c)} }\cdot X^d \cdot Y^e \cdot dX \cdot dY\right]$$ Homework Equations a,b,c are constants; d & e are non negative integers; X and Y are variables. F is a one to one function. Please simplify. The...
  23. M

    How to set up double integral for volume by rotation

    Homework Statement Find the volume of the solid of revolution of the area bounded by the curves about the given axis. y= x^2 - 2, y = 0, about y = -1, consider only the area above y=-1 Homework Equations The Attempt at a Solution So I drew out the problem and figured it would be...
  24. M

    Double Integral over a triangular region

    Homework Statement Let f(x,y) = sin(∏*y^2). Let R be the triangular region on the x-y plane with the vertices at (0,0) (0,1) (.5,1). Consider the solid that is under z = f(x,y) and over the region R. Calculate the volume over that region using double integrals. Homework Equations The...
  25. A

    Solving Double Integral with Polar Coordinates

    Hi all. Suppose that we want to compute the following indefinite integral: The correct solution by Mathematica: Now here is the (apparently) incorrect solution by using polar coordinates: \iint\frac{1}{\sqrt{x^2+y^2}}dxdy=\iint\frac{1}{r}rdrd\theta=(r+c_1)(\theta+c_2) If...
  26. N

    Double integral and reversing order

    Homework Statement http://img10.imageshack.us/img10/3390/55486934.jpg Homework Equations This is what I was thinking: tan−1(∏x)−tan−1(x)=∫^{g(x)}_{f(x)}h(y)dy The Attempt at a Solution I don't really understand how to do this question
  27. M

    How to draw double integral functions

    Hi every one! I would like to draw a double integral function in related to R and h parameters by below M-File but It does goes wrong! Is there anyone to correct it for me? thank you syms R h; a1 = 0; a2 = atan(R./(R+h)); r1 = h; r2 = sqrt(R.^2+(R+h).^2); integrand =...
  28. T

    Multivariate Taylor expansion or else a double integral identity

    Homework Statement This is part of a larger problem, but in order to take what I believe is the first step, I need to take the Taylor series expansion of f(x,y) = \cos\sqrt{x+y} about (x,y) = (0,0) On the other hand, the purpose of doing this expansion is to find an asymptotic expression for...
  29. L

    Double Integral of a Quarter Circle: Evaluating (x^2+y^2)arctan(y/x)

    Homework Statement Evaluate the integral (x^2+y^2)arctan(y/x) for 0<y<a and 0<x<(a^2-y^2)^0.5. Homework Equations The Attempt at a Solution I tried changing the order of integration to get the integral (x^2+y^2)arctan(y/x) for 0<x<a and 0<y<(a^2-x^2)^0.5. I noticed that this was a quarter of...
  30. Z

    Double integral over triangle with known nodes

    I am wondering is there any general analytic solutions to the following integrals First moment of area Integral(xdxdy) Integral(ydxdy) Second moment of area) Integral(x^2dxdy) Integral(y^2dxdy) Integral(xydxdy) Over the triangle (x1,y1) (x2,y2) (x3,y3) thanks Wei
  31. L

    Double Integral (underneath a surface and above a square)

    Homework Statement The volume underneath the surface z= y/ (1+xy) and above the square {(x,y)| 2≤x≤3 , 3≤ y≤ 4} is:Homework Equations Please see attachment. The Attempt at a Solution Please see attachment for solution. My professor had provided us with 8 possible solutions (where only one...
  32. DryRun

    Double integral using transformations

    Homework Statement http://s2.ipicture.ru/uploads/20120109/dT4m6rNG.jpg The attempt at a solution x=\frac{u}{1+v} and y=\frac{uv}{1+v} Transforming the integrand: \frac{x+y}{x^2}e^{x+y}=\frac{(1+v)^2 e^u}{u} dxdy=J.dudv J=\frac{v(1+v)^2 +1+uv}{(1+v)^3} The double integral becomes: \int\int...
  33. DryRun

    Evaluate double integral using transformations

    Homework Statement http://s2.ipicture.ru/uploads/20120107/vVVkUT7f.jpg The attempt at a solution I plotted the graph x-y: http://s2.ipicture.ru/uploads/20120107/ja3V9aSV.jpg y=\frac{1}{2}(u+v) and x=\frac{1}{2}(u-v) So, after finding the Jacobian, the double integral becomes: \int\int...
  34. Q

    Solving Double Integral: ∫∫ sin(x+y) dxdy

    Homework Statement find ∫∫ sin(x+y) dxdy in the domain D where D=(x,y) where y≥√x and y≤2x and y≥(1/x) and y≤2/x Homework Equations i The Attempt at a Solution took y as variable so i have two domains D1 where x is between 0.5 and 1 and y between 1/x and √x D2 where x is between 1...
  35. D

    Double Integral volumes: Triangular base

    Evaluate ∫∫R 5x2 + 2y2 where R is triangle (1,1) (2,0) (2,2) I see the lines bounding the triangle are y=x y=2-x and x=2, and have tried many attempts at setting up the correct limits. Would it be correct to split this into 2 triangles, or are the limits y=x∫y=2-x for y and 2∫1...
  36. DryRun

    Find area of region (double integral)

    Homework Statement Find area of region bounded by x^2 + y^2 = 4above y=1 The attempt at a solution OK, so i drew the graph. http://s1.ipicture.ru/uploads/20111226/oR99IdxJ.jpg The red part of the graph is the area that i need to find. Put y = 1 in the equation of circle; x^2 = 3 x = -√3 and...
  37. DryRun

    Double Integral Evaluation: Why Do We Integrate with Respect to y First?

    Homework Statement Evaluate the double integral: http://s1.ipicture.ru/uploads/20111224/K3MGcdS7.jpg The attempt at a solution Consider x as constant, first integrate w.r.t.y. I get \frac{xy^2}{2} After evaluating the integral with limits y=0 to y=y, i get the same exact answer. Then, i...
  38. T

    Change of variables in a double integral

    Homework Statement Find the mass of the plane region R in the first quadrant of the xy plane that is bounded by the hyperbolas xy=1, xy=2, x^2-y^2 = 3, x^2-y^2 = 5 where the density at the point x,y is \rho(x,y) = x^2 + y^2. Homework Equations The Attempt at a Solution The...
  39. K

    How to evaluate this double integral?

    Homework Statement ∫0 to 2 ∫x/4 to 1/2 (sin (pi*y2)) dy dx Homework Equations The Attempt at a Solution I think I have to convert this to polar or do some sort of change of variable. Although in polar y = r sin θ, so then you would have sin of a sin??
  40. K

    Change to polars double integral

    Homework Statement Here is the question Homework Equations The Attempt at a Solution So converting to polars by x = r cos θ, y = r sin θ gives me r^5 in the inner integral but how do you convert the ranges? y goes from 0 to sqrt 1 - x^2, so that is just r, I could tell just by looking...
  41. T

    Convert the double integral to polar coordinates

    Homework Statement Evaluate the double integral by converting to polar coordinates. ∫∫ arctan y/x dA; R is the sector in the first quadrant between the circles 1/4= x^2+y^2 and x^2+y^2=1 and the lines y=x/√3 and y=x. Homework Equations arctan y/x= θ The Attempt at a Solution...
  42. A

    Double Integral Limits for Triangular Region

    Homework Statement function inside is (x+y)^2 * sin(x^2 - y^2) R is the triangular region w/ vertices (0,0) , (0,2) , (1,1) x = (u+v)/2 y = (v-u)/2 What are the correct limits ?? The Attempt at a Solution Also, when plugging in x and y in the function, i ended up getting...
  43. A

    Double Integral Problem: How to Evaluate ∫1 to 4∫0 to y(2/(x^2+y^2))dxdy

    Homework Statement Evaluate: ∫1 to 4∫0 to y(2/(x^2+y^2))dxdy Homework Equations The Attempt at a Solution So I know you have to spilt it up and do the dx integral first: ∫0-y(2/(x^2+y^2))dx Now this is where I don't know if I'm doing it right, I moved the 2 outside the...
  44. P

    Double integral over a region bounded by an ellipse

    Homework Statement Evaluate. ∫∫D x2 dAxy, bounded by 5x2 + 4xy + y2 = 1 Homework Equations ∫∫D H(x,y) dAxy = ∫∫D H(u,v)\frac{\delta(x,y)}{\delta(u,v)}dAuv The Attempt at a Solution So I understand I'm supposed to find a change of variables to transform the ellipse into a circle...
  45. N

    Double Integral Polar Coordinates to find area of region

    I'm studying for my final and tutors/my professor isn't available over the weekend. Could someone please spend a little time to help me? My problem is stated as: Let R be the right half of the circle x2+(y-1)2=1. Use a double integral polar coordinates to find the area of the region R. I...
  46. K

    Double integral absolute value.

    Homework Statement I just need to know if what i did is correct. The question is as follows: http://imgur.com/1RL7e Homework Equations The Attempt at a Solution What I did is as follows: I split this integral into two parts and solved. [integral from 0 to 1...
  47. K

    Double Integral Evaluation: Finding the Overlapping Area

    Homework Statement Evaluate the integral: integral D of y sin (x+y^2) dA where D = [0x2] x [0x2] U [1,3] x [1,3] Homework Equations The Attempt at a Solution So D is basically a square which simplifies to D = [1,2] x [1,2] since that is the portion of both rectangles that overlap. So then...
  48. K

    What are the boundaries for a double integral over a specific region?

    Homework Statement D = {x>0, x^2 < y < 10-x^2) compute integral (integral D of y^2 sqrt x) Homework Equations The Attempt at a Solution I'm having trouble figuring out the bounds of the integral. y goes from x^2 to 10-x^2 but I think I have to split this integral up...
  49. A

    Calculators Definite Double Integral on Wolfram

    How would you solve the following problem on Wolfram|Alpha (www.wolframalpha.com)? Problem If C is a circle or radius 1 with equation x^2+y^2=1, find \iint_{C} (x^2 + y^2) \cdot \mathrm{d}x \cdot \mathrm{d}y
  50. X

    Help with double integral and switching the order of the differentials

    Homework Statement 1 1/2 ∫ ∫ e-x2 dx dy 0 y/2 Homework Equations ***Graph equation*** The Attempt at a Solution I graphed the function and they made a horizontal strip. However, I can't seem to find the right function with a vertical strip, which is where I'm stuck.
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