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. I

    Double integral: Cartesian to Polar coordinates

    Homework Statement ∫∫√(x^2+y^2)dxdy with 0<=y<=1 and -SQRT(y-y^2)<=x<=0 Homework Equations x=rcos(theta) y=rsin(theta) The Attempt at a Solution 0.5<=r=1, we get r=0.5 from -SQRT(y-y^2)<=x by completing the square on the LHS then, 0<=theta<=pi But, when I calculated the...
  2. M

    How do I calculate this double integral using a change of variables?

    The problem is as follows. Calculate the double integral of cos ((x-y)/(x+y)) dA over R, where R is the triangle bounded by the points (0,0), (2,2), and (2 + pi, 2 - pi). I understand that you have to set U = x-y and V = x+y. However, I am having a hard time finding the bounds on the...
  3. M

    Double integral: Two lines and one curve

    My problem: I don't know how to begin with this problem.
  4. M

    Polar coordinates to set up and evaluate double integral

    Homework Statement Use polar coordinates to set up and evaluate the double integral f(x,y) = e-(x2+y2)/2 R: x2+y2≤25, x≥0 The Attempt at a Solution First I just want to make sure I'm understanding this my double integral would be ∫^{\pi/2}_{-\pi/2} because x≥0 ∫^{5}_{0}...
  5. F

    Double Integral Choosing Change of Variable

    Homework Statement Evaluate the following integral with a change of variable of your choice. \int_0^{1} \int_y^{y+2} \sqrt{(x-y)}dxdy The Attempt at a Solution I'm supposed to choose a u and v that will simplify the integral, but I have no idea how to even start this. I tried...
  6. I

    Limits of integration question (double integral)

    so in the image in the link below, i don't understand a couple of things: 1.) the center of the cylinder is off to the side and not at the center. where/how in the problem are we taking this into account? because it should definitely affect the volume under the parabaloid right? 2.) most of...
  7. M

    Double integral to find volume of a solid

    Homework Statement Set up a double integral to find the volume of the solid bounded by the graphs y=4-x2 and z=4-x2 The attempt at a solution I drew myself a 3d graph but it's just a parabloid in the xy plane and a parabloid in the xz plane right? so I'm unsure how to set up my...
  8. A

    Double Integral With probability

    $$f(x,y)= \begin{cases} 4(x+y^2),&\text{if x > 0, y > 0, x + y < 1} \\ 0, &\text{elsewhere} \\ \end{cases}$$ Find two different integral expressions for P(y > x) (1) $$\int_{0} ^{1/2}\int_{0} ^y 4(x+y^2),dx\,dy + \int_{1/2} ^{1}\int_{0} ^{1-y} 4(x+y^2),dx\,dy $$ (2) $$\int_{0} ^{1/2}\int_{x}...
  9. A

    Determining two sets of boundary conditions for a double integral prob

    Homework Statement Determining two sets of boundary conditions for a double integral problem in the polar coordinate system. Is the below correct? Homework Equations The Attempt at a Solution There are two sets of boundary conditions that you can use to solve this problem in the polar...
  10. A

    Forced to use symmetry to solve this double integral?

    Homework Statement http://i.imgur.com/d4ViHux.png Homework Equations The Attempt at a Solution The author writes: "Now, using symmetry, we have..." But what symmetry does the author use? Also, I got the integral as shown in the remark but why is it wrong?
  11. M

    Can This Double Integral Be Solved in Closed Form?

    is there a closed form solution for this double integral? \int^{2}_{1}\int^{3}_{4}\sqrt{1+4x^{2}+4y^{2}}dydx
  12. J

    Can the Difficult Double Integral Be Simplified with Approximate Functions?

    Here is the beast \iint_{(ax+\mu _{1})^{2}+(bx+cy+\mu _{2})^{2}\leqslant z}\frac{1}{2\pi \sigma ^{2}}e^{-(\frac{1}{2\sigma ^{2}})(x^{2}+y^{2})}dxdy The integral gives the C.D.F. of (ax+\mu _{1})^{2}+(bx+cy+\mu _{2})^{2}\leqslant z where x and y are identically distributed gaussian random...
  13. Ed Aboud

    Double integral, cylindrical coordinates

    Homework Statement The problem states: Use cylindrical coordinates to evaluate \iiint_V \sqrt{x^2 +y^2 +z^2} \,dx\,dy\,dz where V is the region bounded by the plane z = 3 and the cone z = \sqrt{x^2 + y^2} Homework Equations x = r cos( \theta ) y = r sin( \theta ) z =...
  14. Philosophaie

    Double Integral Surface Area of Spherical Ball

    Homework Statement Double Integral Surface Area of Spherical Ball radius Homework Equations ##\int_S d\vec{S} = 4*\pi*a^2## The Attempt at a Solution ##\int\int_0^a f(r,?) dr d? = 4*\pi*a^2##
  15. T

    Double integral over a region needing polar coordinates.

    1. Evaluate the double integral ∫∫arctan(y/x) dA by converting to polar coordinates over the Region R= { (x,y) | 1≤x^2+y^2≤4 , 0≤y≤x } My attempt at solving Converting to polar using x=rcosθ and y=rsinθ I get ∫∫arctan(tan(θ))r drdθ I understand that I have to integrate first with respect...
  16. J

    What is the solution to a double integral problem with given limits?

    Homework Statement ∫∫x^4ydxdy x [-5,10] y [-1,1] (don't know how to do a definite integral in the math code...) The answer choices are A)10^5 B)0 C)-10^{10} The attempt at a solution \frac{x^5y}{5} evaluated at -5 to 10. then ∫20625ydy evaluated at -1 to 1. My final answer is 20625. What...
  17. PsychonautQQ

    Use symmetry in double integral.

    Homework Statement Evaluate the double integral of (2+xy^2) over dA (dxdy) using symmetry where R = [0,1] x [-1,1] Homework Equations The Attempt at a Solution I don't know how to use symmetry to evaluate this.. However if I integrate this integral normally i first get...
  18. A

    Double Integral of an absolute value function - Need Help

    Hi! Need help in solving this double integral: 1 1 ∫ ∫ |x-y| dydx 0 0 Thanks in anticipation. Regards, Aby.
  19. L

    Calculating Area with Double Integrals - Solving for Unknown Functions

    "Hey guys, how are you? I was studying for my calculus final and stumbled upon a peculiar function. Homework Statement Now I have to find the area bounded by the function (x^2+y^2)^3=xy^4 using a double integral. Now, the problem is that the graph is totally unknown to me (I have some...
  20. L

    Finding area using double integral

    Hey guys, how are you? I was studying for my calculus final and stumbled upon a peculiar function. Now I have to find the area bounded by the function (x^2+y^2)^3=xy^4 using a double integral. Now, the problem is that the graph is totally unknown to me (I have some ideas but I am not shure). A...
  21. L

    Solving a Mysterious Double Integral: Help Needed!

    Hey guys, how are you? I was studying for my calculus final and stumbled upon a peculiar function. Now I have to find the area bounded by the function (x^2+y^2)^3=xy^4 using a double integral. Now, the problem is that the graph is totally unknown to me (I have some ideas but I am not shure). A...
  22. A

    Double integral over triangular region

    Homework Statement Integrate f(u,v)= v - sqrt(u) over the triangular region cut from the first quadrant by the line u+v=64 in the uv plane. Homework Equations I am assuming u is the equivalent of the x-axis in the xy plane and v the equivalent of y in the xy plane. I am taking the...
  23. FeDeX_LaTeX

    Can anyone help me understand double integrals involving intersecting cylinders?

    Homework Statement Find the volume of the region common to the intersecting cylinders ##x^2 + y^2 = a^2## and ##x^2 + z^2 = a^2##. The Attempt at a Solution I am totally stuck here. What do they mean when they say 'intersecting cylinders'? I've drawn graphs of circles of radius a...
  24. FeDeX_LaTeX

    Volume of Region Bounded by Elliptic Paraboloid & Plane z=0

    Homework Statement Find the volume of the region bounded by the elliptic paraboloid z = 4 - x^2 - \frac{1}{4}y^2 and the plane z = 0. Homework Equations -The Attempt at a Solution I'm not really sure where to start with this. This is how they've set it up: 4 \int_{0}^{2} \int_{0}^{2 \sqrt{4 -...
  25. S

    Finding the volume of a sphere with a double integral

    Homework Statement I know how to find the volume of a sphere just by adding the areas of circles, so I decided to do a double integral to find the same volume, just for fun. Here's what I've set up. I put 8 out front and designed the integrals to find an eighth of a sphere that has its center...
  26. E

    Getting a double integral over a region

    Homework Statement Let B \in ℝ2 the region is bounded by x^2 + y^2 = 4, \ x^2 + y^2 = 1, \ x^2 = y, \ 2x^2 = y evaluate \iint \limits_{B}\frac {2x^2+y^2}{xy}The attempt at a solution I needed to get the limits of integration. I used the following formula to start with: \iint...
  27. G

    Double Integral over Elliptical Area: Polar Coordinates and Substitution Method

    Homework Statement Calculate \int \int x dx dy Over the area defined by 1 \leq x^{2} + 4y^{2} \leq 9 Homework Equations The Attempt at a Solution First we'll do the sub: u = x + y v = sqrt(3)y Which gives us the area 1 \leq u^{2} + v^{2} \leq 9, u,v\geq0 and the integral \sqrt{3} \int...
  28. E

    Integrating a natural log in double integral

    Homework Statement \int^{1}_{0}\int^{e^x}_{e^-x}\frac{lny}{y}dydx The attempt at a solution So I am integrating ln(y)/y and I tried it by parts, first with u = ln(y), dv = 1/y, and therefore du = 1/y, and v = ln y but if I use that I get (ln(y))2-\int\frac{lny}{y} again. So I tried...
  29. T

    MHB Solution to Double Integral Problem

    I want the answer for this and how is it solved. double integral(x2+y2 dxdy) over the region in pos quadrant for which x+y<=1.
  30. A

    Double Integral Help: Solving for e^sin(x) over D

    double integral help?? Homework Statement Evaluate the double integral | | e^sin(x) dA over the region D = {(x,y) | 0 ≤ x ≤ π/2, 0 ≤ y ≤ cosx} . Homework Equations The Attempt at a Solution how would i do this? i know that dA = dy(dx) so the integral would be |(0 ≤ x ≤ π/2) |(0 ≤ y ≤...
  31. G

    Marginal distribution, double integral clarification

    Homework Statement (X,Y) is uniformly distributed over the area T = {(x,y): 0 < x < 2, -x < 2y < 0} Find the marginal probability functions ie f_{x}(x) and f_{y}(y). The Attempt at a Solution The thing I'm having trouble with is that y depends on x. Am I supposed to rewrite the...
  32. M

    Double integral for loop of rose r=cos2θ

    Homework Statement Homework Equations The Attempt at a Solution Solution is given. I don't understand how +-∏/4 is found as a range for θ Also why is 0 <= r <= cos2θ r is always r which is defined as cos2θ
  33. M

    Rule of thumb for double integral order

    Is there any rule of thumb for which variable should be integrated first?, i.e. to make the whole process of double integration easier.
  34. C

    Help with double integral - volume between 2 surfaces

    Homework Statement find volume of the solid bounded by the surfaces z = 1- \sqrt{\frac{x}{4}^2 + \frac{y}{2 sqrt{2}}^2} and x^2/4 -x +(Y^2)/2 = 0 and the planes z = 0 and z = 1 Homework Equations z = 1- \sqrt{\frac{x}{4}^2 + \frac{y}{2 sqrt{2}}^2} and x^2/4 -x +(Y^2)/2 = 0...
  35. Petrus

    MHB Solving a Double Integral with MHB

    Hello MHB, I got problem understanding how they can do this. \int_0^1\int_x^1 \sin(y^2)dydx and rewrite it as \int_0^1\int_0^y \sin(y^2) dxdy What I have done is. Then function is continuous (\sin is a trig function) on a type I region D. We got D= (x,y)| 0\leq x \leq 1, x \leq y \leq 1 Regards,
  36. Petrus

    MHB The double integral of f over rectangle R and midpoint rule for double integrals

    Hello MHB, I wanted to 'challange' myself with solve a problem with midpoint and rule and the double integral f over the rectangle R. This is a problem from midpoint. "Use the Midpoint Rule m=n=2 to estimate the value of the integrab \int\int_r(x-3y^2)dA, where R= {(x,y)| 0\leq x \leq 2, 1 \leq...
  37. T

    Double integral of ((x^3)+1)^(1/2)

    So I have to evaluate the integral from y=0 to y=1 of(the integral from x=(y^(1/2)) to x=1 of ((x^3)+1)^(1/2)dx)dy. I've substituted the ((x^3)+1) with sec^2(u) since I used tan^2(u)=x^3. I'm wondering if this is the correct (or even a good) manner of solving this because I'm ending up with a...
  38. Petrus

    MHB Calculating a Double Integral in the First Quadrant

    Calculate the double integral, where D is the set of all points in the first quadrant which satisfies the inequality . I am confused how to calculate the a,b \int_a^b (I don't know what you call that in english) Shall I do like this y=4-x and put it in the function so we get x^2(4-x)^2 and then?
  39. B

    MHB The Origin of the \hat x Term in Evaluating Double Integrals over Triangles

    Folks, Self reading a book in which an equation is given as I_{mn}\equiv\int_{\Delta} x^m y^n dx dy where we are integrating an expression of the form x^m y^n over an arbirtrary triangle. Is the above actually a double integral because of the dxdy term? Ie can this be written...
  40. C

    Double integral new coordinate system calculation

    Homework Statement This is a 2 part question. I'm fine with the first part but the 2nd part I'm struggling with. The first part asks us to calculate the double integral, \int\intDx2dA for, D = {(x,y)|0≤ x ≤1, x≤ y ≤1} For this part I got an answer of 1/4. For the 2nd part we introduce a new...
  41. D

    Trigonometric substitution in double integral

    Homework Statement Let R = \{ (x,y) \in \mathbb{R^{2}}: 0<x<1, 0<y<1\} be the unit square on the xy-plane. Use the change of variables x = \frac{{\sin u}}{{\cos v}} and y = \frac{{\sin v}}{{\cos u}} to evaluate the integral \iint_R {\frac{1} {{1 - {{(xy)}^2}}}dxdy} Homework Equations...
  42. C

    Evaluating a double integral with an e^(x^2) term

    Homework Statement Evaluate ∫(0-1)∫(sqrt(y)-1) (ye^(x^2))/x^3 dx dy Homework Equations The Attempt at a Solution First, factor out the y for the inner integral, making ∫(0-1) y∫(sqrt(y)-1) (e^(x^2))/x^3 dx dy And evaluating the inner integral first: y∫(sqrt(y)-1)...
  43. A

    Double Integral One Loop of the Rose

    Homework Statement Use a double integral to find the area of one loop of the rose r = cos 3\theta Homework Equations The Attempt at a Solution This is a past test question. The only thing I got wrong was the set up while I got the rest of the mechanical steps right. I set up as...
  44. skate_nerd

    MHB Double integral of general region

    This problem has brought something up that's making my brain wrinkle. It says to find the integral for $$\int_{ }^{ }\int_{D}^{ }xy\,dA$$ where \(D\) is the region bounded by \(y=x\), \(y=2x-2\), \(y=0\). I have to find the \(dx\,dy\) integral and then find the \(dy\,dx\) integral and evaluate...
  45. skate_nerd

    Changing order of integration double integral

    Homework Statement Set up an integral for the ∫∫xydA for the region bounded by y=x, y=2x-2, and y=0. Set up the dxdy integral, then the dydx integral, then evaluate the simplest of the two. Homework Equations The Attempt at a Solution I drew the region which was easy enough, and...
  46. skate_nerd

    MHB Evaluating a double integral in polar coordinates

    I've done this problem and I have a feeling it's incorrect. I've never done a problem like this so I am kind of confused on how else to go about doing it. The goal is to change the cartesian integral $$\int_{-a}^{a}\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}\,dy\,dx$$ into an integral in polar...
  47. A

    Simple Region Question for a Double Integral Substitution

    Homework Statement Evaluate the double integral integral ∫∫2x^2-xy-y^2 dxdy for the region R in the first quadrant bounded by the lines y=-2x+4, y=-2x+7, y=x-2, and y=x+1 using the transformation x=1/3(u+v), y=1/3(-2u+v).Homework Equations The Attempt at a Solution I've obtained the Jacobian...
  48. LunaFly

    Double integral of arctan in polar coordinates

    Homework Statement Evaluate the integral using polar coordinates: ∫∫arctan(y/x) dA Where R={ (x,y) | 1≤ x2 + y2 ≤ 4, 0≤y≤x Homework Equations X=rcos(T) Y=rsin(T) r2=x2 +y2 The Attempt at a Solution First thing was drawing a picture of R, which I think looks like a ring 1 unit thick...
  49. E

    Double Integral set up problem

    Homework Statement (exact wording from my homework set) Set up an iterated integral for the volume of the region which is above the plane z=5 and below the graph of f(x,y)=21-(x^2+y^2)^2. Pay attention to what the region of integration should be! Homework Equations Not sure. The...
  50. A

    Understanding the Lagrangian Function for Maximization Problems

    Hi everyone! I really need help for this. I have to read a paper in economics where some parts I don't understand. Suppose: S \equiv [\alpha,\bar{\alpha}]x[y,\bar{y}] V^e(p_j,g_j,y+r(r\alpha)-T_j,\alpha)\equiv \underset{h}{\text{max}}U(y+r(r,\alpha)-T_j,p_jh,h,g_j;\alpha) And...
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