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. T

    How to prove the double integral definition of logarithm?

    Where does this definition come from: $$\ln n = \int_{0}^{\infty} \int_{1}^{n} e^{-xt} dx dt$$ Thank you very much.
  2. B

    Setting up a Double Integral for Moment of Inertia

    Homework Statement >Problem:<br>Find the Moment of Inertia of a circular disk of uniform density about an axis which passes through the center and makes an angle of $\dfrac{\pi}{6}$ with the plane of the disc. Homework Equations Moment of Inertia ($I$) is $$\int r^2dm$$ where $r$ is the...
  3. I

    Force on a superconducting cube

    Hi everyone, I need some help to look if I did these calculations right.Let us assume a three dimensional magnetic field: ##\vec{B}(x,y,z) = B_x(x,y,z)\hat{x} + B_y(x,y,z)\hat{y} + B_z(x,y,z)\hat{z}## The equation for the force on a superconducting particle in a magnetic field is given by...
  4. vjacheslav

    What is double integral by interpretation?

    Very simple question for you, friends. As is well known, usual integral has interpretation as square under function's graphic. Then, what is double (and triple) integral by analogue? Thanks!
  5. B

    Fundamental theorem of calculus for double integral

    I was reading about double integral when a doubt came to my mind: how to find the antiderivative of the function f(x,y), like bellow, and compute the fundamental theorem of calculus for double integral? \int_{2}^{8} \int_{2}^{6} f(x,y) dx \wedge dy = ? OBS: It's not an exercise. I know how...
  6. kostoglotov

    Gravity of a disk acting on a mass on the z axis

    Homework Statement A lamina has constant density \rho and takes the shape of a disk with center the origin and radius R. Use Newton's Law of Gravitation to show that the magnitude of the force of attraction that the lamina exerts on a body of mass m located at the point (0,0,d) on the positive...
  7. S

    Double integral on triangle using polar coordinates

    Homework Statement Let R be the triangle defined by -xtanα≤y≤xtanα and x≤1 where α is an acute angle sketch the triangle and calculate ∫∫R (x2+y2)dA using polar coordinates hint: the substitution u=tanθ may help you evaluate the integral Homework EquationsThe Attempt at a Solution so the...
  8. kostoglotov

    Multiple Integral Challenge Question, no solution in guide

    I have what I think is a valid solution, but I'm not sure, and when I try to check the answer approximately in Matlab, I don't get a verified value, and I'm not sure if my analytic solution or my approximation method in Matlab is at fault. 1. Homework Statement Evaluate the integral...
  9. I

    Calculating A Double Integral Using Polar Coords

    I'm on a tablet and having trouble with the math symbols so, for clarity, ∫[a,b] xdx is the integral from a to b of x with respect to x, and f(x) |[a,b] is a function of x evaluated from a to b. Problem: ∫[-1,1]∫[-√(1 - y2),√(1 - y2)] ln(x2 + y2 + 1) Relevent Equations: x2 + y2 = r2 ∫udv =...
  10. E

    MHB What is the value of the double integral?

    Hi, I need to evaluate the following double integral. I have tried direct integration but the answer is too complicated for it to be a viable method. First integral is from 0 to (1-y^2) function is (x^2+y^2)dx. Second integral is from 0 to 1 dy. I can't figure out how to use the maths thing...
  11. kostoglotov

    Double Integrals: Where am I making a mistake?

    Homework Statement Find the volume of the solid. Under the paraboloid z = x^2 + y^2 and above the region bounded by y = x^2 and x = y^2 Well, those curves only intersects in the xy-plane at (0,0) and (1,1), and in the first Quadrant, and in that first Quadrant y = sqrt(x), and over that...
  12. nuuskur

    Double Integral Over a Region: Finding Limits of Integration

    Homework Statement \iint\limits_D x{\rm{d}}x{\rm{d}}y where x = \sqrt{2y - y^2}, y = \sqrt{2x - x^2} Homework EquationsThe Attempt at a Solution I have figured out the region in question: But how do I get the limits of integration? Ah, perhaps.. \int_0^1 \int_{1-\sqrt{1-y^2}}^{\sqrt{2y-y^2}}...
  13. beer

    Graphical interpretation of a double integral?

    Hello, I was helping my friend prepare for a calculus exam today - more or less acting as a tutor. He had the following question on his exam review: ∫∫R y2 dA Where R is bounded by the lines x = 2, y = 2x + 4, y = -x - 2I explained to him that R is a triangle formed by all three of those...
  14. bananabandana

    Calculating Flux through Ellipsoid

    Homework Statement Let ## E ## be the ellipsoid: $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$ Let ## S ## be the part of the surface of ## E ## defined by: $$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$ Let F be the vector field defined by $$ F=(-y,x,0)$$ A) Explain why ##...
  15. RJLiberator

    Double integral problem, conceptual help.

    Homework Statement Find the volume: Prism formed by x+z=1, x-z=1, y=2, y=-2, and the yz-plane. Homework EquationsThe Attempt at a Solution Okay, so I sketched the drawing and I found that I could take the upper region of the xy-plane with respects to x and z and a triangle was formed. The...
  16. RJLiberator

    Help me setup and solve double integral in polar coord.

    Homework Statement The region between sphere x^2+y^2+z^2=3 and the upper sheet of the hyperboloid z^2=x^2+y2+1. Homework EquationsThe Attempt at a Solution Curve of intersection: We set the two equations equal to each other and find x^2+y^2=1, a circle of radius 1 is the curve of...
  17. RJLiberator

    Double Integral of Exponential Function with Changing Bounds

    Homework Statement Double integral of y*e^(x^4-1) with bounds 0=<y=<1 y^(2/3)=<x=<1Homework EquationsThe Attempt at a Solution [/B] Well, the first key thing to recognize is that we need the correct order for the bounds to compute this double integral. So I switch it from x=y^(2/3) and x=1 TO...
  18. A

    Change in Entropy, double integral?

    Homework Statement A thermally conducting, uniform and homogeneous bar of length L, cross section A, density p and specific heat at constant pressure cp is brought to a nonuniform temperature distribution by contact at one end with a hot reservoir at a temperature TH and at the other end with a...
  19. M

    How to Find the Limits of Integration for a Double Integral?

    Homework Statement Evaluate the double integral (x+2y)dA, where R is the region in the first quadrant bounded by the circle x^2+y^2=9. Homework Equations None. The Attempt at a Solution I know how to evaluate the double integral but I just don't know how to find the limits of integration. I...
  20. P

    Leibniz rule for double integrals

    Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...
  21. P

    Double integral in Rectangular coordinates for anything circular

    This is the equation for the cone A \sqrt{x^2 + y^2} The double integral \iint A \sqrt{x^2 + y^2} \space dy \space dx \space \space \space\text {From x= -1 to 1 and y=} -\sqrt{1-x^2} \space to \space \sqrt{1-x^2} \text{ is very difficult to evaluate. I've tried polar coordinate substitution...
  22. A

    Converting single integral to double integral

    Homework Statement Please refer to : http://math.stackexchange.com/questions/1068948/how-to-prove-that-int-0-infty-sinx-arctan-frac1x-mathrm-dx-fra/1069065#1069065 The answer by @venus. What is the procedure in converting that single integral, dividing it into parts, and making it a double...
  23. H

    [Double integral] Area of a triangle

    Hi! I'm stuck with the following problem: ----------------------------------- Calculate ∫∫ (x-y)*|ln(x+2y)| dxdy where D is the triangle with corners in the coordinates (0,0), (1,1) and (-3,3) ----------------------------------- I get the following lines that forms the triangle: y=-x, y=x...
  24. DivergentSpectrum

    Can the Simpsons 3/8 Rule be Extended to Calculate Double Integrals?

    how do i numerically calculate a double integral? as i understand simpsons 3/8 rule is the optimal method for a single integral, is it still true for double integrals? if so, how do i extend the 3/8s rule to do a double integral?
  25. N

    Double integral volume problem

    Homework Statement find the volume of the solid below the plane z = 4x and above the circle x^2 + y^2 = 16 in the xy plane Homework EquationsThe Attempt at a Solution This totally confused me. I didn't think the plane z = 4x sat above the xy plane. If that is true then there would be no solid...
  26. bananabandana

    Two variable function, single integral

    Homework Statement Evaluate: I(y)= \int^{\frac{\pi}{2}}_{0} \frac{1}{y+cos(x)} \ dx if y > 1 Homework EquationsThe Attempt at a Solution I've never seen an integral like this before. I can see it has the form: \int^{a}_{b} f(x,y) dx I clearly can't treat it as one half of an exact...
  27. U

    Double Integral in Polar Coordinates: Evaluating and Solving for Limits

    Homework Statement Evaluate the integral by changing into polar coordinates. \displaystyle \int_0^{4a} \int_{y^2/4a}^y \dfrac{x^2-y^2}{x^2+y^2} dx dy The Attempt at a Solution Substituting x=rcos theta and y=rsin theta , the integrand changes to cos 2 \theta r dr d \theta . I know that the...
  28. P

    Can You Solve These Tricky 2D Integrals on a Unit Circle?

    I can't compute the integral: \int \frac{\arccos(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}\frac{x-a}/{(\sqrt{(x-1)^2+y^2})^3 dxdy on an unit circle: r < 1. for const: a = 0.01, 0.02, ect. up to 1 or 2. I used a polar coordinates, but the values jump dramatically in some places (around the 'a' values)...
  29. W

    Evaluating an Improper Integral using a Double Integral

    Homework Statement Here is a more interesting problem to consider. We want to evaluate the improper integral \intop_{0}^{\infty}\frac{\tan^{-1}(6x)-\tan^{-1}(2x)}{x}dx Do it by rewriting the numerator of the integrand as \intop_{f(x)}^{g(x)}h(y)dy for appropriate f, g, h and then reversing...
  30. J

    Double integral change of variable polar coordinates question

    Homework Statement evaluate the double integral of cosh(6x^2+10y^2) dxdy by making the change of variables x=rcos(theta)/sqrt(3) and y=rsin(theta)/sqrt(5) let D be the region enclosed by the ellipse 3x^2+5y^2=1 and the line x=0 where x>0. Homework EquationsThe Attempt at a Solution first I...
  31. E

    Double Integral for Volume Under a Surface

    Homework Statement Find the volume under the surface z = y(x+2) and over the area bounded by y+x = 1, y = 1 and y = sqrt(x) Homework Equations The Attempt at a Solution Based on the geometry of the bounds, I broke this integral into two parts. I first found the intersection of...
  32. N

    Definite Double Integral of a single variable

    Homework Statement This isn't actually homework. I was messing around in my notebook trying something when I ended up writing something to the effect of this: dT = \frac{V^{2}}{R(1+α dT)}dQ R(1+α dT) dT = V^{2}dQ Where α and V are constants. Now, I'm fairly sure what I had done made...
  33. J

    Evaluating Double Integral of ##\vec{F} \cdot d\vec{s}## on Ellipse

    Homework Statement ##\mathscr{C}## is an ellipse ##\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1## and ##\vec{F}(x,y) = <xy^2, yx^2>## write ##\displaystyle \int_\mathscr{C} \vec{F} \cdot d\vec{s}## as a double integral using greens theorem and evaluate Homework Equations ##\displaystyle...
  34. J

    Average value of a double integral over a region

    Homework Statement f(x,y) = ##e^{x+y}## D is the triangle vertices (0,0), (0,1) , (1,0) Homework Equations ##f(x,y)_{avg}=\frac{\iint_D f(x,y) dA}{\iint_D dA}## The Attempt at a Solution ##\iint_D dA \Rightarrow \int_{0}^{1}\int_{0}^{-y+1} dxdy = \frac{1}{2}## ##\iint_D...
  35. C

    Double Integral of Pythagoras over rectangular region

    Take any given point on the perimeter of a (A x B) rectange and then draw a line from that point to another point on one of the three remaining sides of the rectangle. What is the average length of the line? Well, the answer to that question involves integrals like this: \int_0^A \int_0^B...
  36. M

    Integrating Elliptical Density: A Simplified Approach Using Cross Products

    Homework Statement ∫∫D√(9x2+4y2) dx dyD is the region: x2/4+y2/9=1 My understanding is that i have to integrate the function of a density to calculate the mass of plate which is ellipse. Problem is i can't and shouldn't be able to integrate this integral at my level, so am i missing some way...
  37. V

    Double integral change of variables

    Homework Statement Use the change of variables ##u=x+y## and ##y=uv## to solve: \int_0^1\int_0^{1-x}e^{\frac{y}{x+y}}dydx Homework Equations The Attempt at a Solution So I got as far as: \int\int{}ue^vdvdu. But I just can't find the region of integration in terms of ##u## and ##v##.
  38. K

    Double integral with a circle connecting the two

    I'm trying to figure out what this one symbol was I saw. I also have a guess that I would like to see if is correct. I saw a double integral with a circle connecting the two. What does this mean? Here is my guess. Is it used when dealing with Stoke's Theorem? Since ∫F°dS =∫∫ curl(F)°dS (Both...
  39. D

    MHB Double Integral Problem/ Surface Area of parametric surface

    Hi I'm doing a surface area problem with a parametric surface and I got the cross product but I can't figure out the double integral. I found the solution online but with no explanation, so can someone explain how to solve this integral: thank you!
  40. sa1988

    Is that a bit better?Double Integral in Polar Coordinates

    Homework Statement Homework Equations The Attempt at a Solution As with my other recent posts, I just want to check if I'm right or wrong as I don't have an answer scheme to go by. For this question I simply converted to polar to get: ∫∫(a+a)r drdθ for 0<r<a, 0<θ<2π ...
  41. I

    Use a double integral to find the volume of the indicated solid

    Homework Statement Use a double integral to find the volume of the indicated solid. Homework Equations The Attempt at a Solution I can't find what I did wrong, it seems like a simple problem... $$\int_0^2 \int_0^x (4-y^{2})dydx=\int_0^2 4x-\frac{x^{3}}{3}dx$$...
  42. Digitalism

    Double integral e^(ysqrtx)dxdy

    Homework Statement ∫∫e^(y√x)dxdy from 1 to 4 then from 0 to 2 Homework Equations ∫ e^x = e^x u substitution The Attempt at a Solution I am just curious if this is equal to double integral e^(y\sqrt{x})dydx from 0 to 2 then from 1 to 4. In other words can I change the order of...
  43. Feodalherren

    Moment of inertia, double integral

    Homework Statement Homework Equations The Attempt at a Solution For part B, why is he using the formula for the moment of inertia about the y-axis? Why isn't he using the formula for the moment of inertia about the origin...
  44. J

    Fundamental theorem of calculus for double integral

    The popular fundamental theorem of calculus states that \int_{x_0}^{x_1} \frac{df}{dx}(x)dx = f(x_1)-f(x_0) But I never see this theorem for a dobule integral... The FTC for a univariate function, y'=f'(x), computes the area between f'(x) and the x-axis, delimited by (x0, x1), but given a...
  45. S

    Double integral substitution

    Hey. Homework Statement ∫∫x^3 dxdy, with the area of integration: D={(x,y)∈R^2: 1<=x^2+9y^2<=9, x>=3y} The Attempt at a Solution Did the variable substitution u=x and v=3y so the area of integration became 1<=u^2 + v^2 <=9, u>=v. And the integral became ∫∫(1/3)u^3 dudv. Then I switched to...
  46. JasonHathaway

    Determining double integral limits

    Homework Statement Evaluate \iint\limits_S \vec{A} . \vec{n} ds over the plane x^{2}+y^{2}=16, where \vec{A}=z\vec{i}+x\vec{j}-3y^{2}\vec{k} and S is a part from the plane and R was projected over xz-plane. Homework Equations Surface Integral and Double Integration.The Attempt at a...
  47. JasonHathaway

    Double integral limits from an equation?

    Hi everyone, I've the equation x+y=6 (it's a surface equation which I'll integrate over) and the following integral limits is what I suppose to get it from the equation: \int\limits_0^6 \int\limits_0^{6-x} What's the trick here?
  48. I

    Understanding the Computation of Double Integrals: Can You Help?

    Can anybody please help me understand the computation of the integral in the attached image. I shall be grateful.
  49. applestrudle

    How to Choose Limits for Double Integrals?

    Homework Statement ∫∫ydxdy over the triangle with vertices (-1,0), (0,2), (2,0) Homework Equations I did it like this and got the right answer: ∫dy ∫ydx this first: ∫ydx from x = (y-2)/2 to x = 2-y then ∫dy from y = 0 to y = 2 I got 2 which is correct but when I...
  50. J

    MHB Double Integral Question?

    Find the double integral of (integral sign) (integral sign) ydA where D is the region bounded by (x+1)^2, x=y-y^3, x=-1, and y=-1
Back
Top