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. A

    Evaluate the double integral by converting to polar coordinates

    Homework Statement Convert to polar coordinates to evaluate \int^{2}_{0}\int^{\sqrt(2x-x^2)}_{0}{\sqrt(x^2+y^2)}dydxThe Attempt at a Solution Really I'm just not sure how to convert the limits of integration. I know \sqrt(2x-x^2) is a half-circle with radius 1, but I'm not really sure where...
  2. M

    Volume of liquid in a tank using a double integral

    Homework Statement The fluid level in the tank ((1/4)*(x-4)^2 + y^2 == 4) is 7 m on the left edge of the tank (where x=0) and 5 m on the right edge (where x=8). Find the equation of the plane of the liquid, and use a double integral to find the volume of liquid in the tank. [Hint: you should...
  3. S

    Confused on double integral in polar cords

    Homework Statement Use polar coordinates to find the volume of the solid enclosed by the hyperboloid -x^2-y^2+z^2=1 and the plane z=2. The Attempt at a Solution Solving for z of the equation of the hyperboloid I find z = Sqrt(1 + x^2 + y^2). Letting z = 2 to determine the curve of...
  4. V

    Double integral ( and checking)

    Homework Statement Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy. Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables. Homework Equations The Attempt at a Solution Ok...
  5. V

    What is a Suitable Transformation for a Double Integral on a Parallelogram?

    Homework Statement Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy. Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables.Homework Equations The Attempt at a Solution Ok here is what I have...
  6. S

    Double integral, polar coordinates

    Homework Statement Evaluate \int\intT (x^2+y^2) dA, where T is the triangle with the vertices (0,0)(1,0)(1,1) Homework Equations The Attempt at a Solution \int d\theta \int r^3 dr Thats how far I got, not really sure about boundries on r. First integrals boundrie should be 0 to pi/4. Is...
  7. N

    Double Integral Laws: Moving & Changing Order

    \int_{0}^{\infty}fdx\int_{\frac{x-tx}{t}}^{\infty}dy=\int_{0}^{\infty}dx\int_{\frac{x-tx}{t}}^{\infty}fdy f is a function of x and y can i move f like i showed? can i change the order of integration ?
  8. 8

    Double integral transforming into polar coordinates

    Homework Statement By transforming to polar coordinates, evaluate the following: \int^{a}_{-a}\int^{\sqrt{}{{a^2}-{x^2}}}_{-\sqrt{{a^2}-{x^2}}}dydx Homework Equations The Attempt at a Solution I can get the right answer to this but only after guessing that the inner limits...
  9. 8

    Another double integral problem

    Homework Statement sketch the region of integration, and evaluate the integral by choosing the best order of integration \int^{8}_{0}\int^{2}_{x^{1/3}}\frac{dydx}{y^{4}+1} Homework Equations integration by parts The Attempt at a Solution after sketching the graph and changing the...
  10. E

    Integrating Fresnel Functions (Double Integral)

    Homework Statement All that is provided can be found through the following link: http://img33.imageshack.us/img33/6343/question2q.jpg Homework Equations No specific equations pertaining to solving double integrals. The Attempt at a Solution Ok, so I know that we cannot...
  11. E

    Property of a Double Integral involving a limit

    Hi, I am actually not really concerned about what the whole details are but more whether my approach is correct to show the following statement: Let f be continuous on a closed bounded region \Omega and let (x_0 ,y_0) be a point in the interior of \D_r. Let D_r be the closed disk with center...
  12. R

    Computing a double integral with given vertices

    1. Homework Statement [/b] Use the transformation that takes the unit square to a triangle to compute the integral \int\int_{B}2x+3y dA Where B is a triangular region with vertices (0,0), (5,2), and (3,4). The Attempt at a Solution What I did was I drew the region on an xy...
  13. G

    Evaluate double integral by changing to polar coordinates

    what'd I do wrong? I was told I didn't include the bound y<=x but that still hasn't helped me figure out where I miss stepped thanks -Ben
  14. C

    Is Polar Conversion the Best Approach for This Double Integral Problem?

    Homework Statement http://img23.imageshack.us/img23/3118/intx.th.jpg Homework Equations I'm guessing polar conversion? http://en.wikipedia.org/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates The Attempt at a Solution I'm having trouble tackling...
  15. Saladsamurai

    Evaluating a Simple Double Integral: x+z over x+y+z=1 in the 1st Octant

    Homework Statement I don't know what is going on on my brain. I am at a sage in a problem where I need to evaluate the double integral: \int\int_S(x+z)\,dS where the surface is the is the portion of the plane x+y+x=1 that lies in the 1st octant.The Attempt at a Solution Forging ahead I...
  16. R

    Optimizing Double Integrals with Base e

    \int_0^1\int_0^y e^{x^2} dx dy The region I am integrating over should look like this graph, right? I tried switching the bounds but I am left where what I started. since 0 < x < y, and 0 < y < 1 I can switch to 0 < x < 1 , and x < y < 1 leaving me with the integral...
  17. P

    Double integral change of variable

    Homework Statement Hey all. The problem is to solve the double integral xy da where the constraints C is x^2 + y^2 = 1, with the change of variables x = u^2 - v^2 and y = 2uv The problem is applying the change of variables to the constraint unit circle. After the algebra I end up with...
  18. H

    Calculate the following double integral

    Homework Statement Calculate the double integral: \iint\limits_D x^{5}y^{6}dxdy where D = {(x,y): x9 ≤ y ≤ x1/9} Homework Equations The Attempt at a Solution I didn't think this problem would be too hard, but it seems I'm really not good with double integrals. Anyway, I...
  19. T

    Solving a Double Integral: Where is the Error?

    Homework Statement \int_{D}\int y^2 where D = {(x,y) | -1 \leq y \leq1, -y-2\leq x\leq y The integral I set up is below : \int^{1}_{-1} \int^{y}_{-y-2} y^2 dx dy From that I get the answer 0, but the book says its 4/3. I get 0 because It reduces to this integral ...
  20. C

    Mastering Double Integrals: Solving Tricky Problems with 1/(1-xy) Function

    Homework Statement integral of 1/(1-xy)dxdy x's from 0 to 1 and y's from 0 to 1 The Attempt at a Solution ok so the first integral gives -ln|1-y|/(y) after we evaluated the x's from 0 to 1 but I am having trouble with integrating with respect to y .
  21. V

    Impossible trick question double integral

    Homework Statement Evaluate the double integral sin(x-y)*e(x-y)^2-0y) 2--- dA where D is a disk of radius 2 whose center is (1; 1) Homework Equations The Attempt at a Solution gee this...
  22. V

    Double Integral Over General Region

    Homework Statement 1. Find the volume of the solid which is under the surface z = 2x + y2 and above the region bounded by x = y^2 and x = y^3. Homework Equations The Attempt at a Solution So first I graphed x=y^3 and x=y^2. (http://h.imagehost.org/view/0716/Math_Problem ) I found their...
  23. K

    Help double integral over a region question

    I've tried this question with many different ways and i always got -11.576, but the autograder always marked it wrong. so hopefully i really did something wrong and you can teach me about it. find the double integral of -3*x*y - 3*y over the region bounded by x^2 + y^2 = 9, y = 3x, and y = 0...
  24. J

    How Do I Set Up Limits for a Double Integral Over a Triangular Region?

    Homework Statement I'm supposed to solve a definite double integral. It's supposed to be in the area of the triangle with vertexes at (0,0), (1,1),(0,2) Homework Equations integral of e^(y^2) * dy*dx The Attempt at a Solution First, I need to know the limits of x and y... So, that...
  25. M

    Double Integral in Rectangular Coordinates

    Homework Statement Homework Equations n/a The Attempt at a Solution I set up the intgral at integral from 0 to 5 of integral from 0 to 5y of 8e^(y^2)dxdy I solved it as an iterated integral so I solved the first part, then ended up with integral from 0 to 5 of 40ye^(y^2)...
  26. Z

    Use double integral to find the volume

    Homework Statement bounded by x^2+y^2=r^2 and y^2 +z^2=r^2 i guess r is just a random constant Homework Equations The Attempt at a Solution i don't even have a clue of how to start this question
  27. R

    Changing a double integral to polar coordinates

    Homework Statement Rewrite by converting to polar coordinates, carefully drawing R. \int^{2}_{0}\int^{\sqrt{2x-x^2}}_{0}\sqrt{x^2+y^2}dydxHomework Equations The Attempt at a Solution I believe I have the inside part of it right. What I did was replace the x^2 and y^2 in \sqrt{x^2+y^2} with...
  28. J

    Finding the Area Between Parabolas: Double Integral Help Needed

    I need to find the area between the parabolas x=y^2 and x=2y-y^2, I know I need to use a double integral but am having difficulty finding the limits. Can anyone help please?
  29. M

    Cartesian to Polar in Double Integral

    Homework Statement Solve: \iint_{\frac{x^2}{a^2}+\frac{y^2}{b^2} \leq 1} \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} dx dy Homework Equations Cartesian to Polar The Attempt at a Solution Well - this Integral should be solved as a polar function (the radical should be...
  30. Nabeshin

    How can I evaluate this double integral numerically?

    I have come across the following integral which I need to compute: \int_0^{t_1} \int_{\nu_0}^{\infty} \left(\frac{h \nu ^3}{c^2}\right) \frac{1}{e^{\frac{h\nu}{k T(t)}}-1} d\nu dt The problem is, since the inner integral cannot be computed analytically, I have to resort to numerical...
  31. jinksys

    Finding the Bounds of Theta for a Double Integral

    Example: Use a double integral to find the area of the region: One loop of the rose r = Cos[3 theta] Finding the bounds of r is easy, 0 to Cos[3x]. However, I usually get the bounds of theta wrong. How do I find the bounds of theta without using a graphing calculator and guessing. The...
  32. G

    Converting double integral to polar coordinates

    Homework Statement \int\int(rsin2\vartheta)drd\vartheta sorry i don't see how to put the bounds in but they are 0<\vartheta<\pi/2 and 0<r<2acos\vartheta Homework Equations I know that r=sin\varthetaThe Attempt at a Solution Im really not sure where to start my text is terrible. I really...
  33. S

    Unit Circle Double Integral: Is 2π/3 the Answer?

    For the double integral \int\int_R sqrt(x^2+y^2) dx dy where R is the unit circle. I got\int_0^\pi\int_1^1 sqrt(r2) r dr dtheta Then after the integration I got an answer of 2pi/3 as my final answer. Is this right. The bottom of the 2nd integral is -1 not 1
  34. S

    How do I evaluate double integral as the limit of a sum

    How do I evaluate double integral as the limit of a sum: \int\int 1 dA with a snowflake region constructed as follows: Step 1: Start with a square of area 1 unit2. Step 2: Divide each edge into 3 and construct a smaller square on the middle third, thus creating new edges. Step 3: Repeat step 2...
  35. T

    Improper double integral over R2

    Homework Statement Recall that the integral from -∞ to +∞ of e^(-x^2) is equal to the square root of Pi. Use this fact to calculate the double integral of e^-(x^2 + (x-y)^2 + y^2) dx over the entire region R2. Homework Equations The Attempt at a Solution I am not sure if it's...
  36. S

    Calculating mass by double integral

    Homework Statement The distribution of mass on the hemispherical shell z=(R2 - x2 -y2)1/2 is given by \sigma= (\sigma0/R2)*(x2+y2) where \sigma0 is constant. Find an expression in terms of \sigma0 and R for the total mass of the shell Homework Equations The mass is given by double...
  37. O

    Double integral bounded by closed parametric curve

    question: how do i find the area under f[x,y] bounded by a closed parametric curve x[t],y[t]? it doesn't look like i can use a change of variables. it seems as though double integrals only with functions where the curve is given explicitly such as y[x] or x[y].
  38. B

    What do the sample points mean in a double integral problem?

    Homework Statement I am getting rather confused when I attempt to solve one of these double integral problems. A typical problem is phrased like this: If R = [-1, 3][3,5], use a Riemann sum with m = 4, n = 2 to estimate the value of the following \int\int(y^{2}-2x^{2} The problem will...
  39. T

    Evaluate a simple double integral

    Evaluate a "simple" double integral Homework Statement Evaluate the double integral of f(x,y) = square root (1 - x^2 - y^2) over the disk centred at the origin of radius 1 Homework Equations The Attempt at a Solution So the disc of radius one has boundaries x^2 + y^2 = 1 i am...
  40. T

    Prove that a double integral exists

    Homework Statement Let f(x,y) = 1 if x = 1/3 and y is rational, and let f(x,y) = 0 otherwise. Show that the double integral of f over the region Q = [0,1]x[0,1] in R2 exists (SSQ f dA exists) yet the integral from 0 to 1 of f(1/3, y) does not exist. (sorry for the weird way of writing, I'm...
  41. R

    Double integral variable separation

    \int^{B}______________{A}\int^{\infty}_______________{0}\frac{t^{N-1}x^{s-N-1}dtdx}{e^{t+x}+1} With the restrictions that that B>A, 0<Re(s)<1 and N is a natural number>1. I think t=ab and x=a(1-b) would work, but I'm not sure how to go from there. I don't need to solve the integral; just...
  42. Y

    Change of variables in double integral

    \int_{c_1}^{c_2} \int_{g_1 (x)}^{g_2 (x)} f(x,y) dy dx If f(x,y) is function such that it is not easily integrable, if we wanted to switch the bounds of integration so that h1(y) = g1(x) , same for g2(x), what would be the general way to rewrite the bounds? Would it involve inverse...
  43. R

    Calculating Volume Using Double Integrals: Finding the Boundaries and Limits

    Homework Statement given two surfaces S1={(x,y,z)|z=50-X^2} S2={(x,y,z)|z=9y^2+16} find the volume 1.V1 bounded above by S1 and below by S2 and on the sides by the vertical planes X=1 X=-1 Y=1 Y=-1 2 the solid V2 bounded above by S1 and below by S2 and on the sides by the vertical...
  44. S

    Double Integral - Volume question

    Homework Statement Find the volume of the solid bounded by z = 0 and z = 2xy, lying in the first quadrant and bounded by the curves y = x^2 and x+y = 2 Homework Equations The Attempt at a Solution I have an answer, but just asking if I've done it correctly, since we arent given the...
  45. L

    Double integral for area evaluation

    Homework Statement Use an appropriate double integral and the substitution y = br\sin \theta \text{\ \ \ } x = ar\cos \theta to calculate the bounded area inside the curve: {\left( \frac{x^2}{a^2} + \frac{y^2}{b^2} \right)}^2 = \frac{x^2}{a^2} - \frac{y^2}{b^2} (you can...
  46. P

    Double integral using the dirac delta

    Homework Statement Need to integrate using the dirac delta substitution: \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\!x^2\cos(xy)\sqrt{1-k^2\sin^2(y)}\, dx\, dy Homework Equations \cos(xy) = \frac{1}{2}\left(e^{ixy} + e^{-ixy}\right) \delta\left[g(t)\right] =...
  47. J

    ?Evaluating Double Integral Using Polar Coords

    Homework Statement Use polar coords to evaluate the double integral x3 + xy2dydx from y = -(9-x2)1/2 to (9-x2)1/2, and x = 0 to 3 Homework Equations The Attempt at a Solution So the region is a half circle of radius 3, centered @ the origin, with only the possitive x side...
  48. T

    Cant understand this double integral

    http://www.freeimagehosting.net/image.php?6417d9b089.gif i don't know what it means? what it actually does ? how to explain its function in words ??
  49. T

    Why double integral could calculate area and volume

    why there are a case where double integral could calculate area and in other case it could calculate a volume. an integral should do only one thing not both?? for what characteristics it could used to calculate area, for what its volume
  50. Y

    Changing bounds of integration of a double integral

    Picture of the problem is listed above. I'm not sure how to switch the bounds of integration on it. Anyone here know how to go about this? i tried doing it x^2 to 1 for y and then 0 to 1 for x but it didnt work out to be the write answer, the write answer after putting it in your calculator...
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