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.

View More On Wikipedia.org
Recent contents Top users
  1. F

    Double integral polar/cylindrical coordinates

    Homework Statement Problem 1: Use double integrals to find the volume of the solid obtained by the rotation of the region: ##\triangle = \left\{ (x, y, z) | x^2 \le z \le 6 - x, 0 \le x \le 2, y = 0 \right\} ## (edit) in the xz-plane about the z axis Homework Equations Volume = ##\int_a^b...
  2. F

    Double integral to find area

    Homework Statement Use double integrals to find the areas of the region bounded by ##x = 2 - y^2## and ##x = y^2## Homework Equations Volume = ##\int_a^b \int_{f(x)}^{g(x)} h(x) dx dy##.. and this is equivalent if I switched the integrals and redid the limits of integration The Attempt at a...
  3. karush

    MHB 15.3.65 Rewriting double integral to infnty

    \begin{align*}\displaystyle \int_{\alpha}^{\beta}\int_{a}^{\infty} g(r,\theta) \, rdr\theta =\lim_{b \to \infty} \int_{\alpha}^{\beta}\int_{a}^{b}g(r,\theta)rdrd\theta \end{align*} $\textit{Evaluate the Given}$ \begin{align*}\displaystyle &=\iint\limits_{R} e^{-x^2-y^2} \, dA \\ (r,\theta) \, 2...
  4. lfdahl

    MHB Double integral challenge

    Evaluate the double integral: \[I = \int \int _R\frac{1}{(1+x^2)y}dxdy\] - where $R$ is the region in the upper half plane between the two curves: $2x^4+y^4+ y = 2$ and $x^4 + 8y^4+y = 1$.
  5. H

    Faraday's law with Calc 3 integration help

    Homework Statement Suppose an infinitely long wire carrying current ##I=sin_0(\omega t)## is a distance ##a## away from a equilateral triangular circuit with resistance ##R## in the same plane as shown in the figure. Each side of the circuit is length ##b##. I need to find the induced voltage...
  6. karush

    MHB 15.3.50 Double integral of circle and graph

    $\displaystyle \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \sqrt{x^2+y^2} \, dydx=\frac{\pi}{6}$ this was the W|A answer but how ? also supposed to graph this but didn't know the input for desmos
  7. karush

    MHB 232.q1.2c Double integral with absolute value in integrand

    $\displaystyle \int_{-1}^{1} \int_{-2}^{3}(1-|x|) \,dy\,dx$ ok i was ? about the abs
  8. karush

    MHB Double Integral 232.q1.5a Calculation

    $\tiny{232.q1.5,a}$ \begin{align*}\displaystyle I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\ R&=[0,2]\times[-1,1] \end{align*} would this be $$\int_{-1}^{1} \int_{0}^{2}xy\sqrt{x^2+y^2} \,dx \, \, dy $$
  9. D

    Trying to find this double integral using polar coordinates

    Homework Statement question : find the value of \iint_D \frac{x}{(x^2 + y^2)}dxdy domain : 0≤x≤1,x2≤y≤x Homework Equations The Attempt at a Solution so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x and i decided to convert...
  10. karush

    MHB How do we evaluate the double integral with $15x^2$ inside and all the R values?

    ok just seeing if I have this set up correctly before evaluate.. where does $15x^2$ come from? if $15x^2$ is inside this why would we need all the R values
  11. karush

    MHB 15.2.78 But it asks for a double integral

    Use double integral to compute the area of the region bounded by $y=4+4\sin{x}$ and $y=4-4\sin{x}$ on the interval $\left[0,\pi\right]$ ok it looks easier to do this in one $\int$ but it asks for a double $\int\int$ so ?
  12. karush

    MHB Double Integral: Evaluating $II_{5a}$ in $R=[0,2] \times [-1,1]$

    $\textsf{a. Evaluate :}$ \begin{align*}\displaystyle R&=[0,2] \times [-1,1]\\ II_{5a}&=\iint\limits_{R}xy\sqrt{x^2+y^2}\, dA \end{align*} next step? $$\displaystyle\int_0^1 \int_{-1}^1 xy\sqrt{x^2+y^2}\, dxdy$$
  13. karush

    MHB Evaluating Double Integral $II_{5d}$

    $\textsf{d. Evaluate :}\\$ \begin{align*}\displaystyle II_{5d}&=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{1}{(x^2+1)(y^2+1)} \, dy dx \end{align*}
  14. karush

    MHB 15.1.25 Evaluate the following double integral over the region R

    $\tiny 15.1.25$ $\textsf{Evaluate the following double integral over the region R}\\$ $\textit{note: the R actually is supposed be under both Integrals don't know the LaTEX for it}$ \begin{align*}\displaystyle \int_R\int&=5(x^5 - y^5)^2 dA\\ R&=[(x,y): 0 \le x \le 1, \, -1 \le y \le -1]...
  15. Dirickby

    Double Integral: solution with hypergeometric function?

    Homework Statement Hello, I've recently encountered this double integral $$\int_0^1 dv \int_0^1 dw \frac{(vw)^n(1-v)^m}{(1-vw)^\alpha} $$ with ## \Re(n),\Re(m) \geq 0## and ##\alpha = 1,2,3##. Homework Equations I use Table of Integrals, Series and Products by Gradshteyn & Ryzhik as a...
  16. E

    Understanding Odd and Even Functions in Double Integrals

    Homework Statement Hi, I am not asking for solution for any problem as i already have the given solution for the problem. Instead, what i want clarify is what do they mean by the odd and even function and how do they get 0? Also, is there a need to change the order from dxdy to dydx?
  17. T

    Integrating with respect to area? Past paper question

    This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me). The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...
  18. F

    Double Integral: Error in Calculation?

    Homework Statement I have calculate my double integral using wolfram alpha , but i get the ans = 312.5 , but according to the book , the ans is = 0 , which part of my working is wrong Homework EquationsThe Attempt at a Solution Or is it z =0 , ? i have tried z = 0 , but still didnt get the...
  19. O

    Double integral, find volume of solid

    Homework Statement Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders: y = 1 − x2, y = x2 − 1 and the planes: x + y + z = 2 4x + 5y − z + 20 = 0 Homework Equations ∫∫f(x,y) dA The Attempt at a Solution So I solved for z in the plane...
  20. C

    Double integral to compute capacitance

    Hi everyone. I have quite a basic doubt, and I thought you could help me. Consider the figure: The cylinders S1 is held at a constant potential, and the same applies for the ring identified by S0. All the surroundings are filled with an insulator material. I want to calculate che capacitance...
  21. D

    Issue with Double Integral

    Homework Statement Find the volume of the given solid: Under the surface z = xy and above the triangle with verticies (1,1), (4,1) and (1,.2) Homework Equations Double Integral The Attempt at a Solution I drew the triangle, and found the the equations of the lines to be: x = 1; y = 1; y = -3x...
  22. Drakkith

    Setting up a Double Integral in Polar Coordinates

    Homework Statement Consider the 'ice cream cone' bounded by z = 14 − x2 − y2 and z = x2 + y2 .(a) Find the equation of the intersection of the two surfaces in terms of x and y. (b) Set up the integral in polar coordinates. Homework EquationsThe Attempt at a Solution I got part a without...
  23. nysnacc

    Double integral coordinate transform

    Homework Statement Homework Equations transformation The Attempt at a Solution u = x-y v = x+y I convert each side in terms of u, v, get: u = 0, u = -2 v = 2, v = 4 Correct?
  24. Pull and Twist

    MHB ANSWER CHECK: Double Integral in Polar

    Here is the problem I am dealing with... And this is how I approached it. Can anyone confirm that I did it correctly and got the right answer? Thank you.
  25. T

    I Integration Limits Changing in Double Integral Order Change

    For part of a proof of a differential equations equivalence, we needed to use that $$\int_0^t [\int_0^s g(\tau,\phi(\tau))\space d\tau]\space ds = \int_0^t [\int_\tau^t ds]\space g(\tau,\phi(\tau))\space d\tau$$ I understand that the order is being changed to integrate with respect to s first...
  26. I

    Double integral of gaussian times mod cos function

    Homework Statement I want to evaluate the following definite integral of the form, I = \int\limits_{x = -\infty}^{\infty}\int\limits_{y = -\infty}^{\infty} e^{-ax^2} e^{-by^2} | \cos(c x + d y)| dx dy where a, b, c, and d are constants, as part of a larger problem I am doing, Homework...
  27. N

    Double integral with absolute value

    Homework Statement I am trying to evaluate double integral ∫∫D (|y - x2|)½ D: -1<x<1, 0<y<2 Homework Equations None The Attempt at a Solution I know that in order to integrate with the absolute value I have to split the integral into two parts: y>x^2−−−>√y−x2 y>x^2−−−>√y−x2 I just can't...
  28. C

    Circled Part Formula in Double Integral: Explaining the Use of dA in Polar Form?

    Homework Statement can someone explain about the formula of the circled part? Why dA will become r(dr)(dθ)? Homework EquationsThe Attempt at a Solution A = pi(r^2) dA will become 2(pi)(r)(dr) ? why did 2(pi) didnt appear in the equation ?
  29. H

    I Change of variables in double integral

    I get two different answers, ##a^2## and ##0.5a^2##, by using two different methods. Which is the correct answer? The family of curve for ##y^2=4u(u-x)## is given by the blue curves, and that for ##y^2=4v(v+x)## is given by the red curves. Method 1: Evaluate the integral ##I## directly in...
  30. W

    Another simple double integral

    Homework Statement So i think i got this straight since my last question. let's see :) So my area of integration is: y=4 ; y=x2 and y=(x-2)2 the function is |x-1| i must integrate with respect to dx first. The Attempt at a Solution So i sketched the area (see attatchment graphs should be cross...
  31. W

    Double Integral: ∫∫D x^2 + y^2 dA for D limited by: y=x^2, x=2, y=1

    Homework Statement ∫∫D x2+y2dA where D is the region limited by: y=x2, x=2, y=1 (dA = dxdy) Homework EquationsThe Attempt at a Solution So basically i sketched the area, and i get the area under y=x2 0<x<1 and a square at 1<x<2 , 0<y<1 So i divded the integrals; for the square ∫01∫12 of...
  32. B

    MHB Polar coordinates to evaluate double integral

    I am trying to evaluate \int\int xy dxdy over the region R that is defined by r=sin(2theta), from 0<theta<pi/2. I am struggling on where to begin with this. I have tried converting to polar coordinates but am not really getting anywhere. Any guidance would be really appreciated (Crying)
  33. S

    Double Integral in polar coordinates

    Homework Statement Evaluate ∫∫D (3x + 4y 2 ) dA, where D = {(x, y) : y ≥ 0, 1 ≤ x 2 + y 2 ≤ 4} with the use of polar coordinates. Homework Equations The Attempt at a Solution I made a sketch of the circle. It's radius is = 1 and it's lowest point is at (0,0), highest at (0,2), leftmost point...
  34. B

    What is the volume of a perfect cylinder under a plane?

    Homework Statement Hi ! :) I'm having some difficulties with the question below, in which there are numerous steps and I am unsure in which part(/s!) I have gone wrong. The question is as below; you must via integration calculate the shaded volume of a perfect cylinder of radius R and height...
  35. P

    Moment of inertia avoid double integral?

    Homework Statement Determine the moment of inertia of the shaded area about the x-axis. Homework Equations I(x)= y^2dA The Attempt at a Solution In order to determine the moment of inertia of the shaded area about the x-axis I first looked at the portion above the x-axis, integrate it with...
  36. P

    Moment of inertia (double integral)

    Homework Statement Determine the moment of inertia of the shaded area about the x axis.[/B]Homework Equations Ix=y^2dA The Attempt at a Solution Okey so I now get how to do this the standard method. But I want to know if the method I tried is correct as well or where my mistake lies. My...
  37. Destroxia

    Polar Double Integral Between 2 Regions

    Homework Statement Find the area in the first quadrant that is inside the circle ##r=100sin(\theta)## and outside the leminscate ##r^2=200cos(2\theta)##. I have graphed the region as I interpreted it below. The area I am trying to find is the non-shaded, white region. Homework Equations...
  38. Ekramul Towsif

    Integrating the Integrand: Solving a Double Integral

    Homework Statement ##\int_{z=0}^5 \int_{x=0}^4 \Big( \frac{xz}{ \sqrt{16-x^2}} +x \Big)dxdz## Homework Equations double integration The Attempt at a Solution how do i integrate the term ##\frac{xz}{ \sqrt{16-x^2}}## though i know that ##\int x \, dx = \frac{x^2}{2}## pls help me thoroughly :(
  39. Lucas94

    MATLAB How to do a double integral on matlab?

    Hey! I want to do a double integral calculation of this problem##∫∫ xy/(xy^2 +1)^2## over the region bounded by 2 ≤ x ≤ 3 and 2*sqrt(1+x) ≤ y ≤ 2*sqrt(2+4x) on MATLAB and i have tried the following syntax: clc clear all fun=@(x,y) x*y./((x*y.^2+1).^2); ymax=@(x) 2*sqrt(2+4*x)...
  40. D

    Calc III Double Integral Question

    This is the problem I'm trying to solve. The directions require me to rewrite as a single integral and evaluate. But I'm having trouble setting the bounds for a complete compounded integral. The graph of the region would look something like this... Where the shaded area is the region. I...
  41. P

    Area of z^2=xy inside Hemisphere: Surface Integrals

    Homework Statement Find the area of the part of z^2=xy that lies inside the hemisphere x^2+y^2+z^2=1, z>0 Homework Equations da= double integral sqrt(1+(dz/dx)^2+(dz/dy)^2))dxdy The Attempt at a Solution (dz/dx)^2=y/2x (dz/dy)^2=x/2y => double integral (x+y)(sqrt(2xy)^-1/5) dxdy Now I'm...
  42. W

    Double Integral in Polar Coordinates Symmetry Issue

    Homework Statement Find the volume of the solid lying inside both the sphere x^2 + y^2 + z^2 = 4a^2 and the cylinder x^2 + y^2 = 2ay above the xy plane. Homework Equations Polar coordinates: r^2 = x^2 + y^2 x = r\cos(\theta) y = r\sin(\theta) The Attempt at a Solution So I tried this...
  43. C

    Finding Area with Double Integrals: What is the Approach for This Homework?

    Homework Statement Homework EquationsThe Attempt at a Solution here is my approach, I take the whole area, which is π16 then subtract the unshaded region now to find the unshaded region's area, I use rectangular coordinates. my bounds are from -2 to 2 for x and the the top and bottom of...
  44. H

    Problem integrating a double integral

    Hi, could you please help with the integration of this equation: $$\int_{x}\int_{y}\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\,dydx$$ where ##u(x,y)## . From what I remember, you first perform the inner integral i.e. ##\int_{y}\frac{\partial}{\partial...
  45. T

    The quarter disk in the first quadrant bounded by x^2+y^2=4

    Find the coordinate of center of mass. Given: The quarter disk in the first quadrant bounded by x^2+y^2=4 I tried to solve this problem but can't figure out how to do it. so y integration limit is: 0 <= y <= sqrt(4-x^2)) x limit of integration: 0 <= x <= 2 and then after the dy integral I...
  46. qq545282501

    Volume bounded by two surfaces, what am I missing?

    Homework Statement Find the volume of the solid bounded by z=x^2+y^2 and z=8-x^2-y^2 Homework Equations use double integral dydx the textbook divided the volume into 4 parts, The Attempt at a Solution [/B] f(x)= 8-x^2-y^2-(x^2+y^2)= 4-x^2-y^2 i use wolfram and got 8 pi, the correct...
  47. T

    Double integral to find volume between two surfaces

    Homework Statement set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations y = 4 - x^2 z= 4 - x^2 first octant The Attempt at a Solution I am fairly confident in my ability to evaluate double integrals , but I am having a problem figuring...
  48. C

    Setting up the double integral

    A sheet of metal in the shape of a triangle massing 10 kg per square meter is to be spun at an angular velocity of 4 radians per second about some axis perpendicular to the plane of the sheet. The triangle is a right triangle with both short sides of length 1 meter. (a) The axis of rotation is...
  49. C

    MHB Setting up double integral

    A sheet of metal in the shape of a triangle massing 10 kg per square meter is to be spun at an angular velocity of 4 radians per second about some axis perpendicular to the plane of the sheet. The triangle is a right triangle with both short sides of length 1 meter. (a) The axis of rotation is...
  50. C

    MHB Using Reimann sum to estimate the value of a double integral

    If R = [−3, 1] × [−2, 0], Use a Riemann sum with m = 4, n = 2 to estimate the value of ∫∫R(y2 − 2x2) dA. Take the sample points to be the upper left corners of the squares. So far, I found the indefinite integral of the function to be y3/3 - 2x3/3 Not sure where to go from here
Back
Top