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. 1

    Surface Area [Double Integral]

    I’m doing a lot of double integrals to find surface area problems, and I don’t think I’m setting them up quite right. For example, “Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 25 inside the cylinder x^2 + y^2 = 9.” I converted the sphere to a function of z: \sqrt{25...
  2. T

    Evaluating A Double Integral over a Rectangle

    Homework Statement Let R be the rectangle bounded by x - y = 0, x - y = 2, x + y = 0, and x + y = 3. Evaluate \int\int(x + y)ex2-y2dA R The Attempt at a Solution First I rewrote the boundaries so that I could graph them more easily. I got y = x, y = x - 2, y= -x, and y = -x + 3. I was going...
  3. L

    Double integral from y to 1 and 0 to 1

    Homework Statement Evaluate the integral \int0 to 1\inty to 1\frac{1}{1+x^4}dxdy The Attempt at a Solution I managed to do the first one, from y to 1, using partial fractions and then some substitution, and I get a huge answer involving some logarithms and arctans that don't simplify...
  4. D

    Double Integral problem What am I suppose to do? Related to polar coordinates.

    The problem and my work is shown in the image below. However, I feel like I did something horrible wrong but I'm not sure where! I'm sorry if my handwriting is illegible. If you're having difficulties please leave a comment and I will not hesitate to type it out as a response. Any...
  5. C

    Double Integral Evaluation Using Polar Coordinates

    Homework Statement Evaluate over the x,y plane: ∫∫e^{-\sqrt{x^{2}+4y^{2}}}dxdy And I know the answer SHOULD be \pi Homework Equations Polar-->rectangular identities maybe? x--> rcos, y--> rsinθ, dxdy--> rdrdθ The Attempt at a Solution I tried using polar coordinates, but it...
  6. D

    Double Integral, where did I go wrong? Related to polar coordinates.

    ∫∫cos(x^2 + y^2)dA, where R is the region that lies above the x-axis within the circle x^2 + y^2 = 9. Answer: .5pi*sin(9) My Work: ∫(0 ->pi) ∫(0 -> 9) cos(r^2) rdrdθ u = r^2 du = 2rdr dr = du/2r .5∫(0 ->pi) ∫(0 -> 9) cos(u) dudθ .5∫(0 ->pi) sin(u)(0 -> 9) dθ .5∫(0 ->pi)...
  7. chexmix

    Double integral problem (pretty basic)

    Good day, all: We recently hit double/triple integrals in my multivariable calculus course and I have found that my integration abilities are, well, *beyond* rusty ... and so the problem below, which is one of the very first on my current problem set, has me stumped. Homework Statement...
  8. A

    Double integral problem for finding volume

    Homework Statement A cylindrical drill with radius r1 is used to bore a hole throught the center of a sphere of radius r2. Find the volume of the ring shaped solid that remains. Homework Equations x=r*cos(theta) y=r*sin(theta) The Attempt at a Solution i know that the boundaries...
  9. A

    Writing a double integral from a graph

    Homework Statement A region R is given. (ill just tell you that it is a triangle, given by lines x = -2, y = 2, and y = x). Decide whether to use polar coordinates or rectangular coordinates and write \int\int f(x,y)dA as an iterated intergal, where f is an arbitrary continuous function...
  10. D

    Question about a double integral.

    The double integral xcosy is bounded by y=0, y=x^2, and x=1. I was able to integrate almost wholly through; however, toward the end I was unsure what to do when i was asked to plug in x^2 into x^2. What do I do?! Here is an image of my work on the white board. Please, if my hand writing is...
  11. M

    Double integral of a Log (natural)

    Homework Statement I wish to find the following integral over the rectangle [-a,a] in u and [-b,b] in v using Mathematica. The constants a and b are positive (and non-zero). The variables x and y are in Reals. A(x,y)=\int_{-a}^a{\int_{-b}^b{\log{\left[(u-x)^2+(v-y)^2\right]}{\rm d}v}{\rm...
  12. T

    Double Integral Setup for Finding Area with Given Bounds

    Homework Statement I have the bounds, 0≤y_{1}≤2, 0≤y_{2}≤1, and 2y_{2}≤y_{1}. I now have a line u=y_{1}-y_{2} and I'm trying to find the area such that y_{2}≥y_{1}-u. The integral comes down to two parts, the first of which I'm stuck on (when 0≤y1≤1). I'm pretty sure I have one way setup...
  13. T

    Double integral, help setting up boundaries

    Homework Statement Find the area enclosed by the circles r = 1 and r = 2cos theta Homework Equations The Attempt at a Solution I thought setting bounds of the inner integral as from 2cos theta to 1 and the outer from -pi/2 to pi/2, though this doesn't seem to give me the correct...
  14. E

    Reversing Order of Integration for Double Integral Problem

    My idea was that the limits are and that the anti-derivative of dy was xlog(1+y^2) but that seems wrong... maybe use these limits instead and start with dx? gives us then we take dy guess, i figured it out eventually with the help of wolfram with the last integration
  15. H

    Double Integral: Find Area of Triangular Vertices

    find the area of the double integral ∫∫x + y (is the triangular vertices (0.0) , (2,2) and (4,0)) how to find the values of x and y.
  16. C

    Change of Variables for Double Integral

    Update: I figured out how to solve the problem. Nevermind. Homework Statement Use a suitable change of variables to evaluate the double integral: \int^{1}_{0}\int^{3-x}_{2x}(y-2x)e^{(x+y)^{3}}dydxHomework Equations \frac{\partial(x,y)}{\partial(u,v)}=( \frac{\partial x}{\partial u}...
  17. H

    Double Integral Help: Reversing Order & Finding Limits

    ∫u=3 and l=0 u= x and l= 0∫ (x2 + y2 )dydx solve by reversing the order of integration. u and l means upper and lower limit. this is a double integral by the way. i don't understand how the limits are found when reversing the order and the idea of diagrams. please help me
  18. L

    Double Integral of a Circle with Limits of Integration

    Homework Statement Evaluate f(x,y)=y2\sqrt{1-x2} over the region x2+y2< 1 Homework Equations The Attempt at a Solution using x limits between -1 & 1 followed by the y limits of 0 & \sqrt{1-x2} \int\inty2\sqrt{1-x2}.dy.dx Evaluating this and multiplying be 2 to get the...
  19. A

    Area in cardioid and outside circle - Using Double Integral

    Area in cardioid and outside circle -- Using Double Integral Homework Statement Find the area inside of the cardioid given by r = 1 + cos\theta and outside of the circle given by r = 3cos\theta. Homework Equations \int\intf(x,y)dA = \int\intf(r,\theta)rdrd\theta not really relevant...
  20. E

    Double Integral in Polar Coordinates

    Homework Statement Evaluate \int\intD(x+2y)dA, where D is the region bounded by the parabolas y=2x2 and y=1+x2Homework Equations dA = r*drd\vartheta r2=x2+y2 The Attempt at a Solution Well, I know I need to put D into polar coordinates, but I'm lost on this...
  21. U

    What is the volume under a sphere and above a plane?

    Homework Statement Find the volume under the sphere x^2+y^2+z^2=r^2 and above the plane z=a, where 0<a<r Homework Equations x^2+y^2+z^2=r^2 is the equation of a sphere with radius r centered at the origin z=a is the equation of a plane with height a parallel to the xy plane V = ∫∫z...
  22. B

    Setting up double integral for polar coordinates and integrating

    Link: http://imageshack.us/photo/my-images/39/18463212.jpg/ This is a very long problem so I drew it to make things simpler. Part a) tells me to set up a double integral in polar coordinates giving the total population of the city. I have the following: 2π...4 ∫...∫ δ(r, θ) r dr...
  23. O

    How to Solve a Double Integral with an Elliptical Region?

    [b]1. Find the area of the ellipse (2x + 5y − 3)^2 + (3x − 7y + 8)^2 < 1 I have no idea what this looks like, and hence I can't figure out the limits. Maybe I could transform it into a more familiar form using a translation and rotation? Please help.
  24. P

    Solving Double Integral Using Stokes Theorem for Curl

    Homework Statement Use stokes theorem to find double integral curlF.dS where S is the part of the sphere x2+y2+z2=5 that lies above plane z=1. F(x,y,z)=x2yzi+yz2j+z3exyk Homework Equations stokes theorem says double integral of curlF.dS = \intC F.dr The Attempt at a Solution...
  25. T

    Changing limits of integration in double integral

    Homework Statement Invert the limits of integration of the following integrals: 1 ) \int_{0}^{4} dx \int_{0}^{x} f(x,y)dy \int_{0}^{2} dx \int_{0}^{\surd (4 - x^2)} f(x,y)dy \int_{0}^{1} dy \int_{y}^{2-y} f(x,y)dx These are 3 different integrals in 3 separate exercises, they're...
  26. S

    Why Do I Get Different Answers When Changing the Order of Integration?

    Homework Statement Solve the following Double Intergral, and show the answer is the same, regardless of which order you integrate. The integral is between the boundaries y=x and y=x^2 Homework Equations \int\int_R (x^2 + 2y)dxdy The Attempt at a Solution So first of all i integrated with...
  27. U

    Change to cartesian double integral to polar coordinates and evaluate

    Homework Statement integrate 1/((1+x^2+y^2)^2) dx dy Both x and y going from 0 to infinity Homework Equations x^2+y^2 =r The Attempt at a Solution After that I get 1/(1+r^2) ^2 Cannot visualize the function, do not know what the limits are. If I could have any help it...
  28. P

    Double Integral Calculation with Variable Substitution

    Homework Statement Calculate the double integral over D \int\int x*ln(2x + y)/y^3 dx dy D is the finite area in the xy-plane within the straight lines 2x + y = 1 2x + y = 3 x = y x = 2y Homework Equations - The Attempt at a Solution I thought it was obvious to make the variable substitution u...
  29. G

    Transforming a Double Integral to a Single Integral

    Homework Statement Use polar coordinates to change the following double integral to a single integral involving only the variable r. Double-Integral( \sqrt{1+(x^{2}+y^{2})^{2} ) The x-y region is x^2 + y^2 = 4 in the first quadrant. 2. The attempt at a solution I got upto this...
  30. P

    Double integral conversion to polar coordinates

    i have no idea how to use the functions on here to ill try my best. \int(upper bound a lower bound 0)\int(upper bound 0 lower bound -sqrt(a2-y2) of the function x2y.dxdy firstly trying to map it out... i think its the quarter circle in the top left quadrant with boundaries 0 to a along...
  31. S

    Double Integral of e^(x^4) HELP

    Homework Statement Integrate Double Integral of e^(x^4), First Bound- a = 3*(sqroot(y)) and b= 2 and 2nd Bound - C = 0 and D = 8? Homework Equations The Attempt at a Solution We have been working on this all day and we have tried changing bounds but we cannot find an solution that...
  32. P

    Double integral to calculate area

    Please ignore this thread (problems with Latex and the question looks like a jumbled mess!)
  33. A

    Solve a double integral given area, xbar and ybar?

    Homework Statement If you know the area of a region with constant density, and you know xbar and ybar, then its possible to compute \int\int ax+by dA for any constant a and b. [Hint: write down the formulas for the center of mass of a region. If A=5 and (xbar,ybar)=(2,3), Compute\int\int...
  34. A

    Double integral to simple integral

    Hello. Can anyone help me, please? R = { (x,y) \in R² | 0 \leq x \leq 1, 0 \leq y\leq 1-x} f is continuous at [0,1] Show that \iint_R f(x+y) dxdy = \int_{[0,1]} u f(u) du
  35. E

    Change of variables double integral

    Homework Statement Use the transformation x= \sqrt{v- u}, y = u + v to evaluate the double integral of f(x, y) = \frac{x}{(x^2 + y)} over the smaller region bounded by y = x^2, y = 4 − x^2, x = 1. Homework Equations The Attempt at a Solution d:={ (x,y)| -\sqrt{2}<x<1 , x^2<y<...
  36. Y

    Double Integral bounded by Circle?

    Double Integral bounded by Circle? Double integral of (2x-y)dA bounded by circle of radius 2, centered at origin I just need to figure out the limits for my integrals... I am basically lost, can someone show me how to break this up. I tried doing what I did with the previous triangle bound...
  37. S

    Double Integral in Polar Coordinates

    Homework Statement Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x2 + y2 = 256 and x2 - 16x + y2 = 0. Homework Equations The Attempt at a Solution Finding the intervals of integration for the polar coordinates. From the...
  38. C

    Evaluating Double Integral: Reversing Order and Simplifying by Parts

    Apologies for not being proficient enough in the use of Latex to write this problem properly I hope it will suffice if I simply describe it: It is the integration of f(x,y)=(sin(y))/(x+y) with respect to x between limits 0 and y which I've found to give ( sin(y) ) ( ln(2y) ) This...
  39. J

    Changing order of integration for a double integral

    I'm reading through a proof (the full theorem statement is at the bottom of the post) in a book on probability and I'm having trouble following a line in the proof. The line reads as follows: \int_{0}^{\infty} \int_{x:g(x)>y} f(x) dx dy = \int_{x:g(x)>0} \int_{0}^{g(x)} dy f(x) dx Where...
  40. S

    The boltzmann transport equation double integral

    hi, i've never posted on here before but would appreciate any help given for this question. The scattering term (4th term) in the Boltzmann transport equation contains a double integral. What are you integrating over in each case and why is this? this may be a simple question but...
  41. A

    Double Integral over General Region : Hass Section 13.2 - Problem 5

    Homework Statement Outer Integral: From zero to one dy Inner Integral: from zero to y^2 dx Function is: 3y^3 * e^(xy) Homework Equations None The Attempt at a Solution Have tried numerous u substitutions on e^(xy), but taking me nowhere. I am clearly doing something wrong...
  42. X

    Problem with polar double integral

    Hello everybody, I am having trouble doing this polar double integral. The problem says.. Find the area of the region.. \frac{1}{2}y^2 \leq x \leq 2y 0 \leq y \leq 8 It is hard for me to come up with the limits of integration. Checking the answer would be easy because I can...
  43. V

    How do you evaluate the integral of arcsin(sin(x)) from 0 to 2pi?

    Homework Statement int (1/(4-r^2)^0.5) dr dx, r=0 to 2sinx, x=0 to 2pi Homework Equations How to continue the integral of x The Attempt at a Solution I'm stuck at int(arcsin(sinx)) dx, x=0 to 2pi
  44. D

    Complicated Definite Double Integral

    I've been working on a problem involving a large meteoroid passing over the Earth and what its gravitational effects would be on the Earth's mantle. I developed an equation for this, and I've worked it down to a certain point, but unfortunately, I'm not sure how to finally solve it. By the way...
  45. S

    Yes, your answer is correct. The integral evaluates to 20.25.

    Evaluate \int\int xy dxdy where D is the triangular region {(x,y) element of R2| x+y <= 3, x >=0, y >= 0} ( You have to work out the limits of the integrals from the region D)the bit i get confused about in these questions are the limits of the integrals So i just want to check my answer. I...
  46. L

    Double Integral with Negative Exponent

    Homework Statement ∫4to5 ∫1to2 (1x + y)−2 dy dx The Attempt at a Solution I am confused about what to do with this negative 2. Any ideas?
  47. L

    Solving Diffuse Light Simulation with Double Integral Equation?

    Recently, I've been working on a program to simulate diffuse light, and I've hit a snag. I need to solve (at least so that a computer can compute L(x) quickly) something of the form: L(x)=T(x)+c\int_0^{l_2}\int_0^{l_1} W(x,u_1,u_2) L(u_1) du_1 du_2 W and T are pretty well behaved, and...
  48. L

    Understanding Symmetry in Double Integrals

    Homework Statement Let D be the triangular domain given by 0\leq y \leq3, (y/3)-1 \leq 1-(y/3). Then \int\int (e-x^{5}e^(sqrt(1+y^2)) Homework Equations The Attempt at a Solution There is a quick way to solve it by breaking apart the double integral and then, apparently the x^5...
  49. F

    Change of variables for double integral problem

    Homework Statement I want to use polar coordinates to integrate 1/sqrt(x^2+y^2) dydx with limits of integration 0 < y < x and 0 < x < 3Homework Equations x=rcosO y=rsinOThe Attempt at a Solution I know that the area being integrated over is the triangle enclosed by y=x and x=3. I have my...
  50. S

    Double integral in polar coordinates problem

    Homework Statement \int_{y=-infinity}^{infinity} \int_{x=-infinity}^{infinity} (x^4+y^4)/(1+x^2+y^2)^4 dx dy Homework Equations i'm not sure what the new limits are after the transformation to polar coordinates and how to solve the integral. The Attempt at a Solution i have my...
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