Triple integral Definition and 321 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. C

    Evaluate the following triple integral

    Homework Statement Evaluate the following triple integral I = \int\int\int_{R}x dv in Cartesian coordinates where R is the finite region bounded by the surfaces z=0, y=x^3, y=8, z=x. Sketch the region R. Here dV is the element of volume. Homework Equations The Attempt at a...
  2. W

    Finding Bounds for a Triple Integral

    Homework Statement Find \int \int \int_{D} xydV, where D is the solid bounded by the coordinate planes, the plane x = 1 and the surface z = 16 - 4x^2 - y^2. Homework Equations The Attempt at a Solution I have no problem with actually performing the integration, but I'm lost on...
  3. D

    Calculating Volume for Non-Intersecting Surfaces Using Triple Integral

    Find the volume of the region above the sphere x^2+y^2+z^2=4 and below the paraboloid z = 6-x^2- y^2. Ok so the first thing i did was to find out if the two surfaces ever intersect by substituting x^2+y^2=6-z into the first equation and solving for z. I got only complex solutions, hence they...
  4. J

    Triple Integral of ysinx: Evaluating Limits and Solving for the Solution

    Homework Statement evaluate ysinx from z= 0 to (1-y2)1/2, y = 0 to 1, x = 0 to pi Homework Equations The Attempt at a Solution heres my work -1/2sinx \int(1-y^2)^1/2 y dy from 0 to 1 = \int1/3*sinx dx from 0 to pi = 2/3 Homework Statement
  5. A

    Triple Integral For Moment Of Inertia

    I have general question which need to be answered before I can understand steps which I have to do. There are: When you are told that a solid is bounded by the coordinate plane and the plane x+10y + 2z = 5, are the limits considered to be 0-1 for x-axis, 0-10 for the y-axis and 0-2 for the...
  6. Y

    Question on triple integral polar

    Find the mass of a solid bounded by x = (4-y2)1/2 y = 0 z = 0 z = 1 + x with density = y i understand how to set it upand transform to polar and how to do it but my teacher said its supposed to be -pi/2 to pi/2 for the integral with respect to theta. shouldn't it be 0 to pi/2 because its...
  7. C

    How Do You Set Up Triple Integrals for Bounded Regions in Calculus?

    Homework Statement Consider a region in the first octant bounded by z=1-x^2 and y = 1-x. Write the integral in the order dx dy dz and dx dz dy. Homework Equations The Attempt at a Solution For the first one: z varies from 0 to 1. y (in terms of z) varies from...1 to 1?? x (in terms of z...
  8. G

    Problem determining p in triple integral

    The question states: Find the center of mass of the solid that is bounded by the hemisphere z = sqrt(21 - x ^2 - y^2) and the plane z = 0 if the density at a point P is directly proportional to the distance from the xy-plane. I know that the integral is setup : m =...
  9. E

    How do I set up a triple integral using cylindrical coordinates?

    Homework Statement http://img3.imageshack.us/img3/7558/47586628.th.jpg Homework Equations The Attempt at a Solution I am quite confused whether I should use cartesian, cylindrical, and spherical coordinate.. how do I approach this problem
  10. K

    General question about double and triple integral

    i am not using the template because it doesn't really apply to my question. can anyone explain to me when to use double integral or triple integral for volume? what clues should i look for in the question? also i am still unsure when are polar coordinates used as opposed to cylindrical and...
  11. E

    What is the Correct Function for E in a Triple Integral?

    Homework Statement http://img5.imageshack.us/img5/5222/53026504.th.jpg Homework Equations The Attempt at a Solution I know A-F except for what E is here, I answered sqrt(x^2+y^2) but it is wrong, so what is it supposed to be?
  12. E

    Triple Integral Homework: Negative & Positive Answers?

    Homework Statement http://img5.imageshack.us/img5/6596/67023499.th.jpg http://img5.imageshack.us/img5/3875/13930604.jpg Homework Equations The Attempt at a Solution my answer to the first one is negative and the second one is positive, is this correct?
  13. E

    What Is the Volume of the Object Described by These Triple Integral Limits?

    Homework Statement http://img12.imageshack.us/img12/7181/integral.th.jpg Homework Equations The Attempt at a Solution Well my first attempt is to convert this to a cylindrical coordinate first, which I believed to be: \int_0^1 \int_0^{2\pi} \int_0^1 1 \, dr \,d\theta \,dz is this correct?
  14. E

    Did I Make a Mistake in Computing This Triple Integral?

    Homework Statement http://img19.imageshack.us/img19/5192/captureonr.th.jpg Can anyone tell me if I did any mis-computation on evaluating the triple integral above? Homework Equations The Attempt at a Solution
  15. E

    Triple Integral Evaluation: Where Did I Go Wrong?

    Homework Statement http://img19.imageshack.us/img19/2559/triple.th.jpg Homework Equations The Attempt at a Solution I get 16pi/3 (sqrt(2) -1) as the final result, but when I input the answer to the computer, it doesn't accept it. Am I doing a wrong integration/calculation...
  16. E

    What Should Be the Values for E and F in a Triple Integral of a Sphere?

    Homework Statement The figure below shows part of a spherical ball of radius 5 cm. Write an iterated triple integral which represents the volume of this region. http://img19.imageshack.us/img19/9237/sphereu.th.jpg http://img19.imageshack.us/img19/1699/inte.jpg Homework Equations The Attempt...
  17. S

    A triple integral involving deltas

    SOLVEDHomework Statement evaluate the intergralHomework Equationssorry about how this is going to look don't know the language to display nicely and wouldn't take my copy and pasteall integrals are form -infinity to infinity (x^2+32*z^2)*cos(y)*e^(x-4*z) delta(x-1) delta (y-pi) delta(z-.25)...
  18. H

    What is the best approach for solving a tricky triple integral problem?

    Homework Statement A triple integral, with the bounds, from outer to inner: integrate from -1 to 1 with respect to x integrate from 0 to 1-x^2 with respect to y integrate from 0 sqrt (y) with respect to z on the function x^2*y^2*z^2Homework Equations noneThe Attempt at a Solution I know what...
  19. A

    Setting up triple integral in cylindrical coords (looking to check my answer)

    Homework Statement set up an integral in cylindrical coords to compute the volume of the solid S bounded by the sphere x^2+y^2+z^2=12 and the cone 3z^2=x^2+y^2 where z>=0 The Attempt at a Solution i will post my answer here. please let 'I' stand for integral: i get, I[0,2pi]...
  20. S

    How to Simplify the Triple Integral of z^2 Over a Tetrahedron?

    Homework Statement Essentially, do the volume integral of z^2 over the tetrahedron with vetices at (0,0,0) (1,0,0) (0,1,0) (0,0,1) The Attempt at a Solution There seems to be a ton(!) of brute-force algebra involved. Enough to make me question if I'm doing the problem right. I set up the...
  21. J

    Triple integral using cylindrical coordinates

    Homework Statement \int\int_{Q}\int(x^4+2x^2y^2+y^4)dV where Q is the cylindrical solid given by \{(x,y,x)| x^2+y^2 \leq a^2, 0\leqz\leq\frac{1}{\pi}\}Homework Equations When I convert to cylindrical I get f(r,\theta,z) = r^4\cos^2\theta + 2r^4\cos^2\theta\sin^2\theta + r^2\sin^2\theta, but I...
  22. O

    Need help setting up triple integral in spherical coordinates

    Homework Statement Use spherical coordinates to find the volume of the solid bounded above by the sphere with radius 4 and below by the cone z=(x^2 + y^2)^(1/2).Homework Equations All general spherical conversions Cone should be \phi=\pi/4The Attempt at a Solution So far I think the triple...
  23. H

    Help setting up triple integral

    Homework Statement W is the solid bounded by the three coordinate planes and the surface 4x+2y+3z=16, Calculate Mxz=\int\int\int y dV Homework Equations The Attempt at a Solution the surface 4x +2y +3z=16 is a plane that crosses boundries at (4,0,0), (0,8,0) and (0,0,16/3)...
  24. N

    Can a Shift Simplify the Triple Integral of cos(u+v+w)?

    \int \int \int cos(u + v + w)dudvdw (all integrals go from 0 to pi). I've tried using u substitution for each integral but I end up with a huge integral.
  25. D

    Triple Integral: Convert from Cartesian to Cylindrical Coordinates

    Homework Statement This is my last question about triple integrals in cylindrical coordinates. Evaluate the integral by changing to cylindrical coordinates: \int _{-3}^3\int _0^{\sqrt{9-x^2}}\int _0^{9-x^2-y^2}\sqrt{x^2+y^2}dzdydx Homework Equations In cylindrical coordinates...
  26. D

    Triple Integral in Cylindrical Coordinates

    Homework Statement Find the mass and center of mass of the solid S bounded by the paraboloid z=4x^2+4y^2 and the plane z=a\;\;(a>0) if S has constant density K. Homework Equations In cylindrical coordinates, x^2+y^2=r^2. The Attempt at a Solution In order to find the mass, I tried...
  27. D

    Triple Integral in Cylindrical Coordinates

    Homework Statement Evaluate \int \int \int_E x^2 \, dV where E is the solid that lies within the cylinder x^2+y^2=1, above the plane z=0, and below the cone z^2=4x^2+4y^2.Homework Equations In cylindrical coordinates, x^2+y^2=r^2 and x=r\cos{\theta}.The Attempt at a Solution I tried \int...
  28. D

    Triple Integral in Cartesian Coordinates

    Homework Statement Use a triple integral to find the volume of the solid enclosed by the paraboloid x=y^2+z^2 and the plane x=16 Note: The triple integral must be performed in Cartesian coordinates. Homework EquationsThe Attempt at a Solution I calculated the answer numerically using...
  29. D

    Triple Integral in Cylindrical Coordinates

    Revised question is below.
  30. Saladsamurai

    Triple Integral Spherical Coordinates?

    I don't think so since it's not a sphere (disk). I have not learned about cylindrical coordinates and Cartesian is just a pain, so I am assuming I am supposed to use polar or something. Can someone clear up my confusion? \int\int\int_E y\,dV where E lies above the plane z=0, under the plane...
  31. P

    Triple integral to find the gravity inside a solid sphere

    gravity inside a solid sphere Homework Statement I'm having a hard time setting up a triple integral to find the force of gravity inside a solid sphere. I've done a similar proof in physics before with gravity inside a spherical shell, but it only required a single integral. In this problem...
  32. Saladsamurai

    Triple Integral: Having trouble finding my y bounds

    Homework Statement I=\int\int\int_E x^2e^ydV where E is bounded by the parabolic cylinder z=1-y^2 and the planes z=0 x=1 and x=-1 I know that the graph is a parabola that opens downwards and that has symmetry wrt the x-axis. It also stretches along the x-axis toward + and - infinity...
  33. B

    Triple integral finding bounds?

    Homework Statement Integrate the function over the solid given by the figure below (the bounding shapes are planes perpendicular to the x-y plane, a cone centered about the positive z-axis with vertex at the origin, and a sphere centered at the origin), if P=(0,0,5),Q=(0,4,3), and...
  34. B

    Finding bounds on triple integral?

    Homework Statement Integrate the function over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and and contained in a sphere centered at the origin with radius 20 and a cone opening upwards from the origin with top radius 16...
  35. D

    Triple integral and charge density

    Alright guys I am looking for some help with this problem regarding calculating total electric charge in a layer of ions. This layer of ions is bounded between the planes x+2y+2z=4 and x+3y+3z=3, and by the 3 co-ordinate planes. The density of the ions is rises linearly from zero at the outer...
  36. M

    Can Triple Integrals Be Simplified Using Polar Spherical Coordinates?

    let be the integral \int_{R^{3}}d^{3}r F( \vec r . \vec r , \vec r . \vec a , |r| ,|a|) (1) F depends only on the scalar product of vector r=(x,y,z) and its modulus |r| , hence it is invariant under rotation and traslations (since scalar product is invariant under rotation and...
  37. E

    Triple Integral under a Cone: Limits of Integration Verification

    Homework Statement Triple integral of 1+z inside the cone z=2sqrt(x^2+y^2) above the xy plane and bounded by z=6 Homework Equations The Attempt at a Solution when z=6, 6=2sqrt(r^2) so r=3 limits of integration are z=6 to z=2r r=3 to r=-3 theta=2pi to theta=0 Just want to make...
  38. E

    Triple Integral of z in a Wedge: Correcting Limits for y

    Homework Statement Find the triple integral of z where E is bounded by the planes z=0 y=0 x+y=2 and the cylinder z^2+y^2=1 in the first octant. Homework Equations The Attempt at a Solution Just want to make sure that my setup is right. The limits of integration of x are 2 to 0, for z...
  39. Q

    Spherical Coordinates Triple Integral

    I thought this question was elementary... but I apparently know less than I thought I did. Homework Statement Use spherical coordinates to evaluate \iiint_{E} x^{2}+y^{2}+z^{2}dV Where E is the ball x^{2}+y^{2}+z^{2}\leq 16 Homework Equations x^{2}+y^{2}+z^{2}=\rho^{2} The...
  40. K

    Centroid of a Solid (triple integral)

    Homework Statement Find the centroid of the solid: the tetrahedron in the first octant enclosed by the coordinate planes and the plane x+y+z=1. Homework Equations xcenter = \frac{\int\int\int_G x dV}{V} ycenter = \frac{\int\int\int_G y dV}{V} zcenter = \frac{\int\int\int_G z dV}{V} The...
  41. H

    Triple Integral with Exponential and Radical Functions

    Homework Statement Find \int\int\int\sqrt{x^2+y^2+z^2}e^{-x^2-y^2-z^2}dxdydz The limits of integration for all 3 variables are from -infinity to infinity. Homework Equations This one has me completely stumped, so I'm just wondering if someone could push me in the right direction in how...
  42. T

    Triple Integral: Is \[\frac{{x^3 }}{3}\] Right?

    hi everyone the integral is : \[ I = \int\limits_0^1 {\int\limits_0^x {\int\limits_0^y {ydzdydx} } } \] I'm not sure about the answer , but i think it'll be \[ \frac{{x^3 }}{3} \] am i right ? thanks
  43. R

    Triple Integral for Cylindrical Coordinates in a Parabolic Region

    Homework Statement Find the triple integrals \oint\oint\oint_{W}{f(x,y,z)dV: e^{x^{2}+y^{2}+z}, (x^{2}+y^{2}) \leq z \leq {(x^{2}+y^{2}})^{1/2}Homework EquationsThe Attempt at a Solution So I know I need to probably switch to cylindrical coordinates. But I'm getting confused about the limits...
  44. K

    Triple Integral of z over [0, 2*pi] for r [1, 2]

    [SOLVED] Triple integral Homework Statement Calculate the triple integral of z when z [(r-1), sqrt(1-(r-2)^2)], r [1, 2], tetha [0, 2*pi] 2. The attempt at a solution I've tried again and again, and I always get (17/4)*pi, while the answer is pi/2. Is there anything wrong with...
  45. J

    Using a triple integral to find volume

    I'm supposed to find the volume of a solid bound by co-ordinate planes x=0,y=0, z=0 & surface z=1-y-x^2 and am having a lot of difficulty doing so. f(x,y,z) is not given so I am assuming it is one. I figure I should then take the triple integral dzdydx. Then, I made a 2D sketch for the xy plane...
  46. B

    Triple integral over a sphere in rectangular coordinates

    Homework Statement Evaluate the following integral: \iiint \,x\,y\,z\,dV Where the boundaries are given by a sphere in the first octant with radius 2. The question asks for this to be done using rectangular, spherical, and cylindrical coordinates. I did this fairly easily...
  47. B

    How Do You Calculate a Triple Integral with a Parabolic Cylinder and Planes?

    Triple Integral Help :( Can anyone help me with this triple integral problem? I'm sorry I don't know how to post the script properly; I'm a complete newb. It's a surface integral problem- that part is not important- I have to calculate a triple integral where S is the surface of the volume...
  48. M

    Double or triple integral that equals 30

    For my assignment I have to come up with a really complex double or triple integral that equals 30. Would you mind giving me some ideas?
  49. T

    Triple Integral - Volume Question

    Homework Statement So my question is as follows: Find the volume of the solid bounded by z = 3x^2 + 3y^2 and z = 6sqrt(x^2 + y^2) The Attempt at a Solution I drew the graphs of these out, with the z = 3x^2 + 3y^2 being a circular paraboloid w/ vertex at (0,0,0) and extending in...
  50. R

    Question on limits used in triple integral for volume of a sphere

    Homework Statement I am to derive the volume of a sphere using spherical coordinates. I have derived the (correct) jacobian as r^2sin(theta) dr d(theta) d(phi) so its simply a matter of integrating over the correct limits. Homework Equations What I don't get is why we use 2pi to 0 for...
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