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. S

    Spherical and Cylindrical Triple Integral Conversion: Homework Problem

    Homework Statement . Set up and evaluate the iterated triple integral in spherical coordinates equivalent to the following iterated triple integral in cylindrical coordinates :∫∫∫dzrdrdθ(z goes from r to r√3,r goes from 0 to 1, goes from 0 to 2π Homework Equations conversion of...
  2. W

    Triple integral volume of solid (set up)

    Homework Statement I need help setting this integral up in spherical coordinates, the region above the xyplane, inside the sphere x^2+y^2+z^2=2 and outside the cylinder x^2+y^2=2 The Attempt at a Solution \int^{2\pi}_{0} \int^{\pi/2}_{\pi/4} \int^{\sqrt{2}}_{0}...
  3. A

    Evaluating Triple Integrals on a Sphere in the First Octant

    Homework Statement Evaluate triple integral z^2 dxdydz throughout i) the part of the sphere x^2 + y^2 + z^2 = a^2 (first octant) ii)the complete interior of the sphere x^2 + y^2 + z^2 = a^2 (first octant) Homework Equations It is probably good idea to work in spherical coords. z =...
  4. D

    Triple integral in spherical coordinates

    Homework Statement The problem is to calculate the volume of the region contained within a sphere and outside a cone in spherical coordinates. Sphere: x2+y2+z2=16 Cone: z=4-√(x2+y2) Homework Equations I am having difficulty converting the equation of the cone into spherical coordinates...
  5. K

    Help with a triple integral in spherical coordinates

    Homework Statement Use spherical coordinates. Evaluate\int\int\int_{E}(x^{2}+y^{2}) dV where E lies between the spheres x2 + y2 + z2 = 9 and x2 + y2 + z2 = 25. The attempt at a solution I think my problem may be with my boundaries. From the given equations, I work them out to be...
  6. J

    Triple Integral - How to set up limits?

    Homework Statement Find the limits of this region of integration, and write all possible equivalent iterated integrals given combinations of dz, dy, and dx. Homework Equations none that are really 'equations'? The Attempt at a Solution In particular, I'm having trouble with the...
  7. N

    Finding Volume triple integral by change of variables

    Homework Statement ∫∫∫V 9z2 dxdydz, where V is the solid defined by: -1≤x+y+3z≤1, 1≤2y-z≤7, -1≤x+y≤1 Homework Equations The Attempt at a Solution I did this using the bounds, 1/3(-x-y-1)<=z<=(1/3)(-x-y+1), -x-1<=y<=1-x, -3/2<=x<=1/2 but I think the answer is wrong is there a better way to do...
  8. Z

    Trouble understanding meaning of triple integral in spherical coordinates

    Homework Statement Evaluate \iiint\limits_B e^{x^2 + y^2 + z ^2}dV where B is the unit ball. Homework Equations See above. The Attempt at a Solution Does this evaluate the volume of f(x, y, z) within the unit ball (i.e. anything falling outside the unit ball is discarded)...
  9. I

    Need help in solving a triple integral

    Homework Statement This is an integral I came across while reading a book. It is: \int_0^\infty \int_x^\infty \int_x^\infty cos(t^2-u^2)dt du dx I know the solution is: \frac{1}{2}\sqrt{\frac{\pi}{2}} I want to know how it was solved. The Attempt at a Solution I don't know where to start...
  10. T

    Triple integral for cone in cylindrical coordinates.

    Homework Statement Find limits of integration for volume of upside down cone with vertex on origin and base at z=1/sqrt(2). Angle at vertex is pi/2. Do this in cylindrical coordinates. Homework Equations None. The Attempt at a Solution My inner integral conflicts with the books...
  11. S

    Triple Integral Help in Spherical Coordinates

    Hello everyone. I need help regarding putting boundaries in triple integrals using spherical coordinates. The figure attached below is an ice cream cone. I want someone to explain to me how to put the boundaries in the order of dr dz dθ? I had understood this in class but unfortunately i forgot...
  12. M

    Evaluate the triple integral (with spherical coordinates)

    Homework Statement Firstly sorry for my bad english,i have a one question for you(İ try it but i didn't solve it ) Homework Equations The Attempt at a Solution i know problem will be solved spherical coordinates but i don't know how i get angles (interval) theta and fi ...
  13. C

    Integrating Triple Integral: θ, r, z

    Hi everyone. I am trying to integrate the following: \int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ Here's my work: =2\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}r\sqrt{a^{2}-r^{2}}drdθ I use substitution with u=a2-r2 to...
  14. D

    Triple Integral in Cylindrical Coords

    Homework Statement Construct a triple integral in cylindrical coords to find the volume of the cone r=z, where the height (z value) is limited by z=L. Should be in the form => {int[b,a] int[d,c] int[f,e]} (r) {dr dtheta dz} (Sorry for weird formatting above, brackets purely to make terms...
  15. dkotschessaa

    Triple Integral - Volume of Tetrahedron

    Homework Statement Actually, the problem was addressed in a prior post: https://www.physicsforums.com/showthread.php?t=178250 Which is closed.Homework EquationsI would like to know how HallsofIvy (or anyone) arrived at the formula for the tetrahedron given the vertices (1,0,0), (0,2,0)...
  16. R

    How do I find the volume using cylindrical coordinates for the given region?

    I need some help in this The question states find the volume for the region bounded below by z = 3 - 2y and above by z = x^2 + y^2 using cylindrical coordinates . Now i tried to do it but i got stuck with the part where i find the radius . i found r^2 = 3-2r*sin(zeta) :( its hard to...
  17. H

    Checking method and answer of Triple Integral

    The question is: find the volume of the solid bounded above by x^2+y^2+z^2=9, below by z=0 and on the sides by the cylinder x^2+y^2=4. Now rho comes out to be 3. At the very top of the solid, phi is 0 so z=3. So limits of z are 0 (lower limit) and 3 (upper limit). As far as the...
  18. T

    Someone Please Have Mercy and Help Me Set Up This Triple Integral

    Homework Statement Evaluate ∫∫∫E x2 dV where E is the solid that lies within the cylinder x2 + y2 = 1, above the plane z = 0, and below the cone z2 = 4x2 + 4y2. Solve it using cylindrical coordinates. Homework Equations dxdydz = dzdrdθ, and other typical anti-derivative tricks from...
  19. Who Am I

    Rearranging a triple integral - vector calc

    Rearranging a triple integral -- vector calc Homework Statement Change the order of integration in the following integral to dy dz dx. \int^{1}_{0}\int^{2x}_{0}\int ^{x+y}_{0}F(x,y,z) dz dy dx Homework Equations [/B] N/A 3. Attempt at solution Well, the overall thing that I...
  20. H

    Show that a triple integral = pi/4

    Homework Statement Show that \int\int\int \sqrt{x^{2}+y^{2}+z^{2}} e^{-({x^{2}+y^{2}+z^{2}})} dxdydz = \pi/4 where the bounds of x, y, and z are 0 to infinity (The improper integral is defined as the limit of a triple integral over the piece of a solid sphere which lies in the first...
  21. H

    How Can I Find the Volume of a Solid Bounded by Three Coordinate Planes?

    Can someone help me with this?Homework Statement Find the volume V of the solid S bounded by the three coordinate planes, bounded above by the plane x+ y+ z = 2, and bounded below by the plane z = x+ y.Homework Equations x + y + z = 2 z = x + yThe Attempt at a Solution...
  22. C

    Help with problem about triple integral

    hi all , I am newbie . i need your help ... now I am studying a course about calculus exactly (Jame steward book) . so can u help me how to solve a triple integral by J : example : ((x/a)^2 + (y/b)^2 + (z/c)^2)^2 = (x/a)^2 + (y/b)^2 Thank in advance :) p/s : teach me how to do it .. not that...
  23. B

    Triple Integral Cartesian Coordinates

    Ok I have a quick question. I have this problem that is doable with polar coordinates and triple integrals but I was wondering if it would be possible to do this problem in the cartesian coordinate system (odd question I know...). Homework Statement A sprinkler distributes water in a circular...
  24. M

    Evaluate triple integral, involves e -(x 2)

    Evaluate triple integral, involves e**-(x**2) Homework Statement Evaluate the triple integral of e**-(x**2 + 2y**2 + 3z**2), all of the limits are from -infinity to infinity. Homework Equations The Attempt at a Solution I'm not really sure how to do this problem. I know I have to...
  25. T

    Triple integral in spherical coordinates

    I want to check if I'm doing this problem correctly. Homework Statement Region bounded by x^2+y^2=4 and bounded by the surfaces z = 0, and z=\sqrt{9-x^2-y^2}. Set up triple integrals which represent the volume of the solid using spherical coordinates. Homework Equations...
  26. C

    Boundaries of a triple integral

    Homework Statement Let D = { (x,y,z) } such that x^2 + y^2 < 1 and -1 < z < 1 } denote the interior of a cylinder. Compute the triple integral of ( xyz dxdydz ) Homework Equations The Attempt at a Solution Ok so it seems to me that the boundaries should be as follows, -1< x...
  27. K

    Setting up a triple integral with spherical coordinates

    Homework Statement http://img28.imageshack.us/img28/7118/capturenbc.jpg Homework Equations x2 + y2 + z2 = p2 http://img684.imageshack.us/img684/3370/eq0006m.gif The Attempt at a Solution Using the relevant equations I converted the given equation to: ∫∫∫e(p3/2) * p2 *...
  28. Q

    Triple integral and cartesian coordinates

    we all know that triple integral can be solved by either cartesian coordinates , spherical ,or cylindrical coordinates i just need like some advice in knowing when the variable used is constant and when it is not for example : r in cylindrical coordinates can it be constant or not?? because i...
  29. G

    Triple Integral and finding the average

    Solve the Integral in cylindrical coordinates ∫∫∫ dxdydz/(sqrt( x^2 + y^2 + (h-z)^2) B Where B is the Ball with a Radius R around (0,0,0), and the parameter h is greater than R. And then infer the average on that ball B with radius R of the distance opposite to the point (0,0,h). I...
  30. Q

    Can I use polar coordinates in a triple integral?

    if i am being asked to write the domain of integration in a triple integral problem in a cartesian form , may i used polar coordinates to express instead of x and y? thank you
  31. K

    Triple Integral: Volume Between z+x^2+y^2=4 and x^2+y^2+z^2=6

    Homework Statement find the volume between z +x^2 + y^2 = 4 and x^2+y^2+z^2 = 6 Homework Equations The Attempt at a Solution what i did first was solve for the intersection points of z i got 2 and -1. then you get two equations for x^2 + y^2 x^y+y^2 = 5 and x^2+y^2=2...
  32. K

    Triple integral spherical coordinates.

    Homework Statement Here is the question given: Homework Equations The Attempt at a Solution So i set p as x^2 + y^2 + z^2 so p lies in between b and a. But how do i find the restrictions on the two angles, theta and phi?
  33. M

    Need help with triple integral volume question

    Homework Statement A central cylinder of radius 1 is drilled out a sphere of radius 2. Let B be the region inside the sphere but outside the cylinder. Evaluate ∫B 1/x2+y2+z2 dV The Attempt at a Solution Volume = ∫∫∫ 1/x2+y2+z2 dV = ∫∫∫ 1/\rho2 sin\phi\rho2 d\rhod\thetad\phi...
  34. K

    Triple integral & cylindrical coordinates

    Homework Statement When you are doing a triple integral and convert it to cylindrical co ordinates, how do you find the new ranges of integration? I understand the new range of z, if z is between f(x,y) and g(x,y), you just sub in x = r cos θ and y = r sin θ to find the new functions...
  35. A

    Visualizing a Triple Integral Bounded by Planes

    Homework Statement the function is xyz2 V is bounded by y=1-x, z=0, and z=y. The Attempt at a Solution the limits are: x is from -1 to 1 ? y is from 0 to (1-x^2) ? z is from 0 to y ? the question asks for a picture ... how should that look? there are points on (1,0,0)...
  36. P

    Triple Integral Limits Setup Cartesian

    Homework Statement Find volume bounded by: x - z = 0; x + z = 3; y + z = 1; z = y + 1; z = 0 Homework Equations Vol = \int\int\int_{V}dV The Attempt at a Solution I really don't know how to begin this problem because I have trouble visualizing what all those intersection of...
  37. B

    Triple Integral of minimum value

    Homework Statement The problem is to take the triple integral over B of min{x,y,z}dV where B = {x,y,z}: o<x<1 , 0<y<1, and 0<z<1. (These are all less than or equal to, I didn't know the notation). Homework Equations The Attempt at a Solution What I did was break it up into...
  38. T

    Finding Triple Integral: Planes at x1=0, x2=0, x3=2

    Okay so I need to try to find the triple integral for the following. Planes at x1=0, x2=0, x3=2 Area: x3=x12+22, x1≥0, x2≥0 ∫W√(x3-x22dx So I understand that need to find 3 integrals, the first being ∫02 ∫0√x3-2 ∫0√x32-x22 √x3-x22 dx1dx2dx3 I really don't understand how to find...
  39. S

    Triple Integral - Change the order of integration

    Homework Statement \int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx Find 5 equivalent iterated integrals. Homework Equations 0 ≤ z ≤ y 0 ≤ y ≤ x^2 0 ≤ x ≤ 1 The Attempt at a Solution 1) \int^{1}_{0}\int^{√y}_{0}\int^{x^2}_{0} f(x,y,z) dz dx dy I will try...
  40. K

    Bounds for Triple Integral Region over Triangle

    Homework Statement evaluate the integral: int B of z DV where B is the region between the planes: z = x+y, z = 3x+5y and lies over the triangle with vertices (0,0), (0,1), (1,0) Homework Equations The Attempt at a Solution I'm having some trouble trying to figure out the...
  41. S

    Changing the order of integration for a triple integral?

    Homework Statement \int^{1}_{0}\int^{x}_{0}\int^{y}_{0} f(x,y,z)dzdydx I need to write it in terms of dxdydz Homework Equations The Attempt at a Solution I've tried to draw the 3D representation. I don't really know how to change the order, I don't recall my teacher even...
  42. V

    How do I rewrite this triple integral into the form \int\int\int dxdydz?

    Hi!Can anyone please help me out with this question? Appreciate any help,thanks! Rewrite the integral: ∫0<x<1 ∫0<z<1-x2 ∫0<y<1-x dxdzdy into this form: \int\int\int dxdydz How do I change the integrals?Can any kind souls teach me how to sketch the diagram?I can't visualise it >.<...
  43. M

    Volume usingiterated triple integral

    Homework Statement Write an iterated triple integral in the order dzdydx for the volume of the region in the first octant enclosed by the cylinder x2+y2=4 anf the plane z=4. (You do not need to evaluate) Homework Equations The Attempt at a Solution I think I have the right set up...
  44. O

    Volume between plane x+y+z=1 and xy-plane

    Homework Statement Find the volume between the plane x+y+z = 1 and the xy-plane, for x+y\leq2, x\geq0, y\geq0. The Attempt at a Solution First, the plane is above the xy-plane for y < 1-x and below the xy-plane for y > 1-x, so we'll need two integrals. This is how I set them up...
  45. Z

    Triple integral volume problem, volume between 2 paraboloids

    Homework Statement Find the volume of the solid region E bounded by the paraboloids z = 1+x^2+y^2 and z = 4 - 2x^2 - 11y^2 The Attempt at a Solution i set up a triple integral using Cartesian coordinates but was unable to solve it because the limits of integration where very...
  46. D

    How to Set Up a Triple Integral Over a Tetrahedral Region?

    1. The problem statement, all variables and given known data f(x,y,z,) = e^{2x-z} W: x + y + z ≤ 1 x, y, z ≥ 0 Homework Equations The Attempt at a Solution For each domain, could you check it please? This is the only triple integral that's haunting me 0 ≤ x ≤ 1-y-z 0 ≤ y ≤...
  47. T

    Average value of a function over involving triple integral

    Homework Statement We define average value of function over a solid to be f = 1/Volume int int int f(x,y,z) dV So find the average value of the function f(x,y,z) = x^2z+y^2z over the region enclosed by paraboloid z = 1-x^2-y^2 and the plane z = 0 The Attempt at a Solution Actually...
  48. S

    Evaluating Triple Integral of Region E

    Homework Statement Evaluate the triple integral of the region E, where E is the solid w/i the cylinder x^2+y^2=1, above the plane z=0, and below the cone z^2=x^2+y^2. So is the plane z=0 same as the xy-plane? I was doing a homework problem that has Homework Equations I just need...
  49. V

    How do I solve triple integrals using other coordinates?

    Hi everyone, I have problems solving triple integral question like this: Find the volume of the solid bounded below by the cone \varphi=\frac{\pi}{6} and above by the plane z=a. I can do simple triple integration questions,but can some please give me some guidance on how to solve triple...
  50. M

    Triple Integral converting from cylindrical to spherical

    Homework Statement Convert the following integral to an equivalent integral in spherical coordinates. Do NOT evaluate the integral. ∫∫∫ r^3 dz dr dtheta limits of integration pi/4<theta<pi/2 0<r<2 0<z<√(2r-r^2) Homework Equations z=pcos(theta) r^2=x^2 +y^2 p^2=x^2 +y^2...
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