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. Z

    Determine limits of integration in double integral change of variables

    $$h(t)=f(t)*g(t)=\int_0^t f(\tau)g(t-\tau)d\tau=\int_0^t g(\tau)f(t-\tau)d\tau\tag{1}$$ The Laplace transform is $$H(s)=\int_0^\infty h(t)e^{-st}dt=\int_0^\infty\left ( \int_0^t g(\tau)f(t-\tau)d\tau\right )e^{-st}dt\tag{2}$$ The Laplace transforms of $f$ and $g$ are $$F(s)=\int_0^\infty...
  2. chwala

    Solve the problem involving the given double integral

    Ok in my approach i have the lines, starting with the inner integral, $$\int_0^1 xy \cos (x^2y) dx$$ I let ##u =x^2y , u(0)=0, u(1)=y## ... $$\dfrac{1}{2} \int_0^y \cos u du=\left[\dfrac{1}{2} \sin u \right]_0^y= \left[\dfrac{1}{2} \sin (x^2y) \right]_0^1=\left[\dfrac{1}{2} \sin y...
  3. Skaiserollz89

    Symmetry of an Integral of a Dot product

    This homework statement comes from a research paper that was published in SPIE Optical Engineering. The integral $$\int\int_{-\infty}^{\infty}drdr'W(\vec{r})W(\vec{r'}) \vec{r} \cdot \vec{r'}=0$$ is an assumtion that is made via the following statement from the paper : "Since...
  4. Valour549

    Two ways of integration giving different results

    I am trying to do the double integral. And I remembered there's this formula that says if the integrand can be split into products of F(x) and G(y) then we can do each one separately, then take the product of each result. Taken from Stewart's Calculus 9E. So I tried to do the integral two...
  5. N

    A Double integral with infinite limits

    I have the following problem and am almost sure of the answer but can't quite prove it: ##f(y)## is nonnegative, and I know that ##\int_0^{\infty } f(y) \, dy## is finite. I now need to calculate (or simplify) the double integral: $$\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \...
  6. Leo Liu

    How to take the double integral of a data set with respect to time

    Question: Suppose I have a data file for the acceleration of an object after every ## \Delta t_i##, how do I obtain the displacement of it? Context: Integral in a PID loop, although not exactly what I am asking as one is sum of error: $$\int_0^T \int_0^T \ddot {\vec \theta(t)}dtdt$$ the other...
  7. mncyapntsi

    Confused about polar integrals and setting up bounds

    So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds. Any help would be greatly appreciated! Thanks!!
  8. Saracen Rue

    I How to evaluate the enclosed area of this implicit curve?

    The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below: In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
  9. WMDhamnekar

    MHB Approximating the Double Integral using Monte Carlo Method

    Write a program that uses the Monte Carlo method to approximate the double integral $\displaystyle\iint\limits_R e^{xy}dA$ where $R = [-1,1] \times [0, x^2]$. Show the program output for N = 10, 100, 1000, 10000, 100000 and 1000000 random points. My correct answer: My Java program...
  10. A

    Finding the area of a double integral using dxdy instead of dydx

    I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle... So, I was curious how one might...
  11. B

    A Help needed with derivation: solving a complex double integral

    I need help with a derivation of an equation given in a journal paper. My question is related to the third paragraph of this paper: https://doi.org/10.1007/BF00619826. Although it is about fibre coupling my problem is purely mathematical. It is about solving a complex double integral. The...
  12. A

    Question about a double integral region

    Greetings All! I have a problem finding the correct solution at first glance My error was to determine the region of integration , for doing so I had to the intersection between y= sqrt(x) and y=2-x to do so x=(2-x)^2 to find at the end that x=1 or x=5 while graphically we can see that the...
  13. A

    Double integral with polar coordinates

    Greetings! I have the following integral and here is the solution of the book (which I understand perfectly) I have an altenative method I want to apply that does not seems to gives me the final resultMy method which doesn't give me the final results! where is my mistake? thank you!
  14. W

    Volume of solid region double integral

    I sketched this out. With the z=0 and y=0 boundaries, we are looking at ##z \geq 0## and ##y \geq 0## I believe ##0 \leq x \leq 5## because of the boundary of ##y=\sqrt{25-x^2}##. This is my region ##\int_0^5 \int_0^\sqrt{25-x^2} x \, dydx ## ## =\int_0^5 xy \vert_{0}^{\sqrt{25-x^2}} \, dx##...
  15. ?

    Please evaluate this double integral over rectangular bounds

    Summary:: Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine. [Mentor Note -- thread moved from the technical math forums, so no Homework template is shown] Hi, I'm trying to find the answer to the following integral over the rectangle...
  16. Addez123

    Solve p = P(2X <= Y^2) using double integral

    Background information Earlier they've shown that some double integrals can be simulated if it contains pdfs. Ex: $$\int \int cos(xy)e^{-x-y^2} dx dy$$ Can be solved by setting: Exponential distribution $$f(x) = e^{-x}, Exp(1)$$ Normal distribution $$f(y) = e^{-y^2}, N(0, 1/\sqrt 2)$$By knowing...
  17. Leo Liu

    Find the bounds after changing the variables in a double integral

    The answer calculates the integral with ##du## before ##dv## as shown below. However I decided to compute it in the opposite order with different bounds. Here is my work: According to the definitions, $$\begin{cases} u=x+y\\ v=2x-3y \end{cases}$$ First we need to convert the boundaries in xy...
  18. P

    Double Integral via Appropriate Change of Variables

    Summary:: Calculate a double integral via appropriate change of variables in R^2 Suppose I have f(x,y)=sqrt(y^12 + 1). I need to integrate y from (x)^(1/11) to 1 and x from 0 to 1. The inner integral is in y and outer in x. How do I calculate integration(f(x,y)dxdy) ? My Approach: I know that...
  19. D

    What is the key integration technique needed for this double integral?

    Dear all, Last semester on the final exam, our professor gave us an integral that seems difficult to solve. The integral came at the end of a lengthy problem, where we were asked to find the net Gauss curvature of Enneper's surface. The integral that emerged is the following. We tried...
  20. D

    Reversing the order of integration in a double integral

    Performing the x-integration first the limit are x=y2 and x= -y2 and then the y limits are 0 to 1. This gives the final answer 2/5 But i am getting confused when trying to reverse the order of integration. My attempt is that i have to divide the region in 2 equal halfs and then double my answer...
  21. A

    Shortcuts to find a solution to a double integral

    I know the value of this integral is equal to 0, but I would like to see if there is any tricks to spot this answer using symmetries or even odd propreties? Thanks in advance
  22. A

    Double Integral Problem: Incorrect Jacobian Calculation for Polar Coordinates

    calculate the double integral over the region of integration is x^2 + y^2 ≤ 4; x^2 + (y/4)^2 ≥ 1 the integrals have been made over two regions my problem is that when I go to the polar coordinate for the ellipsis and use the jacobian i got 2 instead of 8 ( the following is the professor...
  23. A

    Problem with a double integral

    I already have the solution in which the region of integration has been divided into two regions but I was wondering if I can only use one region considering the polar coordinate system) the disk equation for me is r=2cos(θ) and the theta goes from 0 to (pi/4) 0<r<2cos(θ) and the 0 <θ<pi/4...
  24. tworitdash

    A Spectral domain double integral with singularities

    The integral looks like Y_{mut, mn} = -j^{m+n}nm \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{2 ab (k^2 - k_x^2) \sin^2(\frac{k_yb}{2}) \cos^2(\frac{k_xa}{2})}{\omega \mu k_z (\frac{k_yb}{2})^2 [(n\pi)^2 - (k_xa)^2][(m\pi)^2 - (k_xa)^2]} dk_x dk_y Here, k_z = -1j \sqrt{(-(k_0^2 -...
  25. Zack K

    Verifying the flux transport theorem

    Let ##S_t## be a uniformly expanding hemisphere described by ##x^2+y^2+z^2=(vt)^2, (z\ge0)## I assume by verify they just want me to calculate this for the surface. I guess that ##\textbf{v}=(x/t,y/t,z/t)## because ##v=\frac{\sqrt{x^2+y^2+z^2}}{t}##. The three terms in the parentheses evaluate...
  26. D

    Does it look like I'm doing this double integral correctly?

    are the boundaries of integration correct? i split the domain in two as follows -2<=x<=0 , -(4-x^2)^(1/2)<y<=x+2 and 0<=x<=2 -(4-x^2)^(1/2)<=y<=(4-x^2)^(1/2)
  27. D

    Double integral domain with absolute value

    D={(x,y)∈ℝ2: 2|y|-2≤|x|≤½|y|+1} I am struggling on finding the domain of such function my attempt : first system \begin{cases} x≥2y-2\\ -x≥2y-2\\ x≥-2y-2\\ -x≥-2y-2 \end{cases} second system \begin{cases} x≤y/2+1\\ x≤-y/2+1\\ -x≤y/2+1\\ -x≤-y/2+1\\ \end{cases} i draw the graph and get the...
  28. D

    I Question about this double integral

    could please some one explain the inequality on the right? in particular how should i see and thanks
  29. R

    I Double integral and Green's theorem

    Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region
  30. R

    Change of variable in a double integral

    Hi everyone, I tried to solve the last part of the question, I substituted back the expression of x and y into the equation of the ellipse, I got that r=1 or r=-1. But got no idea how to find the boundary for theta, I got a guess that, It should be from zero to pi. But got no reason why to...
  31. A

    Double integral with polar coordinates

    Hello there, I'm struggling in this problem because i think i can't find the right ##\theta## or ##r## Here's my work: ##\pi/4\leq\theta\leq\pi/2## and ##0\leq r\leq 2\sin\theta## So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta## Which is equal to...
  32. H

    A Double Integral with Dirac Delta Function and Changing Limits

    I have an integral: \int_{-1}^{0}\int_{-1}^{q}\delta(s+a)\sinh[k(q-s)]dsdq where 0<a<1 and \delta (s-a) is a dirac delta function. Anyone know what to do?
  33. M

    MHB How to evaluate a double integral over a bounded region?

    how do I evaluate the double integral 2x + y dA over the region R bounded by y = 3x, 2X + y = 5, x = 0, and y = 0
  34. A

    How to solve a surface double integral?

    Hi I´d like a suggestion about a surface double integral. If I have a sphere x^2+y^2+z^2=4 is on the top of a cardioid r=1-cosθ. The problem is when I solve the integral I got a inverse sine when the answer is a natural logarithm (ln)
  35. D

    How to prove that ##f(x,y)## is not integrable over a square?

    I'm confused with how Riemann sums work on double integrals. I know that ##L=\sum_{i,j}fm_{ij}A_{ij}## and ##U=\sum_{i,j}fM_{ij}A_{ij}## where ##m_{ij}## is the greatest lower bound and ##M_{ij}## is the least uper bound and ##A_{ij}## is the area of each partition. ##A_{ij}=\frac{1}{n^2}## for...
  36. M

    Integrating with Double Limits: X or Y? Calculating the Correct Integral

    So i drew sketch. And I do not understand, how to write integral for calculation, which I should use, X or Y on limit? Is one of them right? First answer gives me 65,7 Second 383,4
  37. Gursimran Singh

    Potential energy of a shell and a disc, both covered uniformly with charge

    Double integration maybe?? I calculated potential due to shell on plate's center but not on other points on it's surface.
  38. G

    Double integral - What are the upper and lower bounds?

    Cant get my head around the order of the upper and lower bounds for this, Is it always the higher take away the lower?
  39. T

    Converting Cartesian to Polar (Double Integral)

    Homework Statement Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy. I am having difficulty finding the bounds for theta and r. Homework Equations I understand that somewhere here, I should be changing to x = r cost y = r sin t I understand that I can solve...
  40. Jazzyrohan

    I Change of order in double integrals

    In the question given below, can we change the order of integral so that y can be the independent variable and x be the dependent one?The cylinder x^2 + z^2 = 1 is cut by the planes y=0,z=0 and x=y.Find the volume of the region in the first octant.This may look like a homework question but it's...
  41. S

    Double Integral: How to Evaluate a Double Integral over a Pentagonal Region

    Homework Statement Evaluate ##\int\int_{R} (x+2)(y+1) \; dx \; dy## where ##R## is the pentagon with vertices ##(\pm 1,0)##, ##(\pm 2,1)## and ##(0,2)##. Homework EquationsThe Attempt at a Solution After drawing ##R## I split ##R## into two sections ##R_1## (left half) and ##R_2## (right half)...
  42. Mr Davis 97

    I Finding new region for double integral

    I have a double integral where the region of integration is ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. I am doing the change of variables ##u=zt## and ##v = z(1-t)##. I am a bit rusty on calc III material, so how would I find the new region of integration, in terms of the...
  43. O

    If you do not answer the above questions, you will not have a correct answer.

    Homework Statement Determine the area of the surface A of that portion of the paraboloid: [x][/2]+[y][/2] -2z = 0 where [x][/2]+[y][/2]≤ 8 and y≥x Homework Equations Area A = ∫∫ dS The Attempt at a Solution Area A = ∫∫ dS = 3∫∫ dS
  44. karush

    MHB 232.5a Evaluate the double integral

    $\tiny{232.5a}\\ \textsf{Evaluate the double integral}$ \begin{align*}\displaystyle I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\ R&=[0,2]\times[-1,1] \end{align*} Ok, just want to see if I made the first step correct. this looks like simply a rectangle so x and y are basically...
  45. J

    A Maximization Problem: Double Int. w/ C not Dependent on Integrals

    Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2} $$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...
  46. S

    Area Calculation for Circle and Cardioid Using Double Integrals

    Homework Statement r=1 and r=1+cos(theta), use a double integral to find the area inside the circle r=1 and outside the cardioid r=1+cos(theta) Homework EquationsThe Attempt at a Solution I am confused on the wording and how to set it up. I tried setting it up by setting theta 0 to pi. and r...
  47. D

    MHB Double integral Problem (with solution)

    Evaluate (use attached figure for depiction) $ \iint_{R} \, xy \, dA $ where $R$ is the region bounded by the line $y = x - 1$ and the parabola $y^2 = 2 x + 6$. I will post solution in just a moment with a reply.
  48. M

    Using Green's Theorem for a quadrilateral

    Homework Statement Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful. Homework Equations Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...
  49. S

    Volume of Double Integral: Finding the Region with Graphed Equations

    Homework Statement z=x^2+xy ,y=3x-x^2,y=x find the volume of the region Homework EquationsThe Attempt at a Solution I graphed y=3x-x^2 and y=x I am confused on which region I use to find the volume. Do I use the upper region or the lower region.
  50. CollinsArg

    Hard Double Integral Homework: Solve & Understand

    Homework Statement I'm given the integral show in the adjunct picture, in the same one there is my attempt at a solution. Homework Equations x = r.cos(Θ) y = r.sin(Θ) dA = r.dr.dΘ The Attempt at a Solution [/B] I tried to do it in polar coordinates, so I substituted x=r.cos(Θ) y=r.sin(Θ) in...
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