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. T

    Help setting up a triple integral

    Homework Statement Hi guys, I need help setting up an integral. Problem: Compute the integral f(x,y,z)=xyz over the solid region bounded below by plane z=-x, above by z=x, and otherwise b the parabolic cylinder x=2-y^2 This is not a surface integral, is it? Because the problems...
  2. M

    Triple integral in spherical coordinates

    Homework Statement use spherical coordinates to calculate the triple integral of f(x,y,z) over the given region. f(x,y,z)= sqrt(x^2+y^2+z^2); x^2+y^2+z^2<=2z The Attempt at a Solution Once I find the bounds, I can do the integral. But I'm having trouble with the bounds of rho. This...
  3. M

    Triple Integral Volume: Octant x,y,z>=0 Bounded by x+y+1=1 and x+y+2z=1

    Homework Statement Find the volume of the solid in the octant x,y,z>=0, bounded by x+y+1=1 and x+y+2z=1 The Attempt at a Solution I've been looking at an example in the textbook that is similar to this problem. First, I found the projection of W onto the xy plane...
  4. M

    Evaluate the triple integral for specified function

    Homework Statement Evaluate the triple integral for specified function and box B. f(x,y,z) = x ey-2z 0<x<2, 0<y, z>1 (The < and >'s should be less than or equal to but, I don't know how to write that here) Homework Equations The Attempt at a Solution I know how to evaluate...
  5. R

    Can Triple Integrals Be Solved in Multiple Ways?

    Homework Statement Homework Equations The Attempt at a Solution
  6. Y

    Triple integral to find Volume of Solid

    Homework Statement Use a triple integral to find the volume of solid enclosed between the sphere and paraboloid. Homework Equations Equation for sphere x2+y2+z2=2a2 Equation for paraboloid az = x2+y2 (a>0) The Attempt at a Solution Trying to find limits of integration: For...
  7. L

    Maximize value that triple integral will compute

    Homework Statement This is a question an exam I already took but I felt as though the solution posted by the professor might be incorrect so I wanted to hear some other opinions. The question is as follows: What is the maximum value that \iiint (-|x|-|y|+1)\,dV will compute for any region...
  8. P

    What are the Correct Bounds for a Triple Integral with Symmetry?

    I'm trying to set up this triple integral with the following bounds: x=0, y=0, z=0, x+y=1, z=x+y. Now I first computed the volume to be 1/3 with a double integral and then what I've been doing is setting what I think are the right bounds for the triple integral and integrating f(x,y,z)=1...
  9. M

    Find the triple integral of xy

    find the triple integral of xy where E is bounded by y = x^2 and x = y^2 and the planes z = 0 and z = x + y. i got 1/3 as a solution, but I'm not sure if i did it right, specifically the part in finding the boundaries for x and y. i found that they intersected at (0,0) and (1,1) so i had the...
  10. E

    Triple integral, 2 parabolic cylinders

    Homework Statement Find the volume of the region which is bounded by the parabolic cylinders y=x², x=y² and z=x+y and z=0 Homework Equations The Attempt at a Solution I solved x=y² for y, and set that equal to y=x², and I got the intersection of the two parabolic cylinders to...
  11. E

    Changing Order of Integration on Triple Integral

    Homework Statement \int^{1}_{-1}\int^{1}_{x^2}\int^{1-y}_{0} dz dy dx Homework Equations See the attachment for graph. I am supposed to rewrite the order of integration to the following. a)dy dz dx b)dy dx dz c)dx dy dz ...and so on. The Attempt at a Solution First attempt is...
  12. N

    Radius of Gyration Triple integral question

    Homework Statement By using spherical coordinates, find the radius of inertia (Is this the same as the radius of gyration?) about the z-axis of the constant density solid which lies above the upper half of the cone x2 + y2 = 3z2 and below the sphere x2 + y2 + (z-2)2 = 4. For a constant...
  13. S

    Finding Volume of Tetrahedron Using Triple Integral

    Homework Statement use triple integral to find the volume of tetrahedron enclosed by the coordinat planes "x=o , y=0 , z=0" and the plane 2x+y+z=0 Homework Equations The Attempt at a Solution I will integrate the constant function f(x,y,z)=1 by the order : dzdydx the...
  14. N

    Very quick triple integral question

    Homework Statement Use the Divergence Theorme to evaluate the flux of v(x,y,z)=x^{2}i+y^{2}j+z^{2}k on the solid T bounded above by a sphere with radius 3 and below by the xy-plane. I've found that div(v) is 2(x+y+z). When I go to set up the integral I get a triple integral over T of...
  15. F

    Triple integral in cylindrical/spherical

    Homework Statement Use cylindrical coordinates to find the volume of the solid. The solid is enclosed by the paraboloid z=x2+y2 and the plane z=9Homework Equations z=r2 The Attempt at a Solution So I'm getting close to the answer but not quite, and I keep getting a negative which doesn't make...
  16. F

    What is the Method for Solving an Easy Triple Integral Problem?

    Homework Statement Use a triple integral to find the volume of the solid. The solid in the first octant bounded by the coordinate planes and the plane 3x+6y+4z=12 Homework Equations z=3-\frac{3}{4}x-\frac{3}{2}y The Attempt at a Solution So I'm using the triple integral...
  17. E

    Triple integral xyz coordinates to spherical

    Homework Statement the integral is bounded below by the cone z^2=x^x+y^2 and above by the sphere x^2+y^2+z^2=18 \int^{3}_{0}\int^{\sqrt{9-y^2}}_{0}\int^{\sqrt{18-x^2-y^2}}_{\sqrt{x^2+y^2}}x^2+y^2+y^2 dzdxdy Homework Equations conversion formulas The Attempt at a Solution I am struggling...
  18. V

    Triple Integral with Spherical Coordinates

    Homework Statement Evaluate \int\int\int 1/\sqrt{x^{2}+y^{2}+z^{2}+3} over boundary B, where B is the ball of radius 2 centered at the origin. Homework Equations Using spherical coordinates: x=psin\Phicos\Theta y=psin\Phisin\Theta z=pcos\Phi Integral limits: dp - [0,2] d\Phi -...
  19. E

    Triple integral to find the volume

    Homework Statement use a triple integral to find the volume of the region that is common to the interiors of z^2 +y^2 + z^2 = 1 and x^2 + z^2 = 1 Homework Equations Would I just calculate the are of the disc? I set up a triple integral as inte [0 to 1] 2nd inte [0 to sqrt(1-z^2)] 3rd...
  20. H

    Triple Integral for Volume in Rectangular Coordinates

    Homework Statement The region R in 3D is cut from the first octant (x,y,z >= 0) by the plane X+Z = 1, Y+2Z = 2. Set up the volume in all 6 ways in rectangular coordinates. Then evalute the volume in two of these ways. Make sure to specify limits of integration in every case. Homework...
  21. K

    Calc 3- Triple Integral using cylindrical coordinates

    Use cylindrical coordinates to evaluate the triple integral , sqrt(x^2+y^2) where the region integrated is the solid bounded by the circular paraboloid z=9-16(x^2+y^2) and the xy-plane. I'm having trouble deciding what the bounds for r would be.
  22. M

    Triple integral - Volume of a sphere and surface

    Homework Statement Find the volume of the solid which is contained by 1) z= \frac{\sqrt{2}}{4}\sqrt{x^2+y^2} and 2) x^2+y^2+z^2= \sqrt{27}z Homework Equations I've completed the square on the 2nd equation to obtain x^2+y^2= 8z^2 and also the 1st equation to obtain...
  23. A

    Prove the improper triple integral equals 2*Pi

    Homework Statement Show that \int^{infinity}_{-infinity}\int^{infinity}_{-infinity}\int^{infinity}_{-infinity}sqrt(x^2+y^2+z^2)e^-^(^x^2^+^y^2^+^z^2^)dxdydz = 2\pi Homework Equations x^2+y^2+z^2 = \rho^2 The Attempt at a Solution I converted to spherical coordinates to get...
  24. J

    Finding the volume with a triple integral

    Homework Statement Use spherical coordinates to find the volume of the solid that lies above the cone z2 = x2 + y2 and below the sphere x2 + y2 + z2 = z Homework Equations I'm going to use { as an integral sign. Volume = {{{ P2 Sin[Φ] dP dΦ dΘ The Attempt at a Solution P2 =...
  25. H

    Triple Integral Using Cylindrical Coordinates

    Homework Statement A conical container with radius 1, height 2 and with its base centred on the ground at the origin contains food. The density of the food at any given point is given by D(r) = a/(z + 1) where a is a constant and z is the height above the base. Using cylindrical polar...
  26. Z

    Finding the Volume using a triple integral

    Homework Statement Find the volume of the solid bounded by the cylinder x^2+y^2=9 and the planes y+z=5 and z=1Homework Equations NoneThe Attempt at a Solution My main problem is setting up the integral. So far what I have is 1 as the integrand, my order of integration is dydxdz and my bounds...
  27. C

    Setting up a triple integral using cylindrical & spherical coordinates

    Homework Statement Inside the sphere x2 + y2 + z2 = R2 and between the planes z = \frac{R}{2} and z = R. Show in cylindrical and spherical coordinates. Homework Equations \iiint\limits_Gr\,dz\,dr\,d\theta \iiint\limits_G\rho^{2}sin\,\theta\,d\rho\,d\phi\,d\theta The Attempt at a...
  28. M

    Calculus Problems, triple integral and polar coordinates stuff

    Hi I have a homework set due this week, 14 problems, I have done 11 of them, but these 3 are giving me trouble, help would be great :) Homework Statement 1.A cylindrical drill with radius 4 is used to bore a hole through the center of a sphere of radius 8. Find the volume of the ring shaped...
  29. M

    Cylindrical Triple Integral: Evaluating Over a Bounded Solid

    Homework Statement Use cylindrical coordinates to evaluate the triple integral (over E) sqrt(x^2+y^2) dV , where E is the solid bounded by the circular paraboloid z=9−16(x^2+y^2) and the xy plane. The Attempt at a Solution This is really bugging me... Is this the correct setup for the...
  30. P

    How do you find the maximum for a triple integral without using a prefix?

    Homework Statement Find the region E for which the triple integral: (triple integral over E) (1 - x^2 -2y^2 -3z^2) dV is a maximum. Homework Equations The Attempt at a Solution I remember in earlier math courses finding the derivative of a single variable integral, does this...
  31. M

    Calculating Volume Using Triple Integrals in Spherical Coordinates

    hi all how can i find the volume of the solid that lies within the sphere x^2+y^2+z^2=36 , above the xy plane, and outside the cone z=7sqrt(x^2+y^2) . your help is very much appreiated
  32. King Tony

    Triple Integral, already solved, need checked

    Homework Statement Let S be the region in the first octant under the plane 3x + 2y +z = 4. Find the volume of S. Homework Equations idk? The Attempt at a Solution \int^{\frac{4}{3}}_{0}\int^{\frac{3}{2}x + 2}_{0}\int^{-3x - 2y + 4}_{0}dzdydx =...
  33. 6

    Converting triple integral coordinates

    [b]1. consider the triple integral (x^2 +Y^2) dV where it is bounded by a solid sphere of radius R. Set up the integral using rectangular coordinatesI tried setting this up with the bounds [ -sqrt(R^2-x^2-Y^2) <= Z <= sqrt(R^2-x^2-Y^2) , -R <= X <= R , -sqrt(R^2-x^2) <= Y <= sqrt(R^2-x^2) ]...
  34. D

    Triple Integral in Rectangular Coordinates Converting to Spherical Coordinates

    Homework Statement Given that: Write an equivalent integral in spherical coordinates. Homework Equations (Triple integral in spherical coordinates.) (Conversions from rectangular to spherical coordinates.)(What spherical coordinates entail) The Attempt at a Solution The region...
  35. K

    How Do You Convert a Triple Integral Into Spherical Coordinates?

    I'm taking a Calculus class as an elective. This might not have been a good idea, but I'm stuck in it now. Here is a problem I have to do. My knowledge of basic maths is poor, so please be gentle and explain thoroughly! 1. The problem. Rewrite the following integral in terms of spherical polar...
  36. V

    Changing evaluation of an axis on a triple integral

    So I'm in the middle of a calculus 3 course, and one thing I've been lightly chewing on is how to change the direction of evaluation of a double/triple integral when the bounds are complicated enough that they can't be drawn easily on a graph. Would you have to use the optimization in several...
  37. K

    HELP Setting up triple integral in spherical coordinate

    HELP! Setting up triple integral in spherical coordinate Homework Statement http://img517.imageshack.us/img517/9139/83291277.jpg Homework Equations I set up the bound for this problem as following: r=0..2/cos(phi), phi=pi/2..3pi/4, theta=0..2pi, but maple always return an error in...
  38. R

    Triple integral and volume - please tell me if i'm wrong

    Homework Statement I need to set up the triple integral to find the volume of the region bounded by the sphere x2 + y2 + z2 = a2 and the ellipsoid \frac{x^2}{4a^2} + \frac{4y^2}{a^2} + \frac{9z^2}{a^2} = 1 Homework Equations The Attempt at a Solution I solved it in spherical...
  39. P

    Triple integral to find volume of ice cream cone

    Homework Statement Use a triple integral in rectangular coordinates to find the volume of the ice cream cone defined as follows The region R in the xy-plane is the circle of radius 1 with center at the origin. The sides of the cone are defined by the function z= \sqrt{x^2+y&2} The top of...
  40. Y

    Setting up a triple integral to find volume of a region

    Homework Statement I need to set up the triple integral to find the volume of the region bounded by the sphere: x^2+y^2+z^2=a^2 and the ellipsoid: (x^2/4a^2)+(4y^2/a^2)+(9z^2/a^2)=1. Homework Equations above The Attempt at a Solution I'm not sure which interval I should be using here. I...
  41. C

    Converting triple integral to spherical

    Homework Statement Evaluate this integral using spherical coordinates: http://img262.imageshack.us/img262/9361/xyzm.th.jpg Homework Equations http://img40.imageshack.us/img40/9508/conss.th.jpg The Attempt at a Solution http://img264.imageshack.us/img264/9457/attempt.th.jpg...
  42. E

    Triple Integral ( IS THIS RIGHT?)

    Triple Integral ( IS THIS RIGHT??) Homework Statement Let R be the solid enclosed by the planes x=0, y=0, z=2 and the surface x^2+y^2, where x\geq0, y\geq0. Compute\int\int\intxdxdydz Homework Equations The Attempt at a Solution I did \int0-1\int0-1\int(x^2+y^2)-2 xdzdydx...
  43. M

    Triple integral with cylindrical coordinates

    Homework Statement Use cylindrical coordinates to evaluate the triple integral \int\int\int \sqrt{x^2+y^2} dV in region E where E is the solid bounded by the circular paraboloid z=9-(x^2+y^2) and the xy-plane. Homework Equations knowing that x = rcos\theta y= rsin\theta z=z...
  44. J

    Is (5*Pi)/6 the Correct Volume for the Intersection of a Paraboloid and Cone?

    Just needing to check an answer really. The question is as follows; The domain bounded by the surface of a paraboloid z=2-x^2-y^2 and that of a cone z^2=x^2+y^2 is given by D = ( x,y,z : x^2+y^2 \leq 1, sqrt(x^2+y^2) \leq z \leq 2-x^2-y^2 ). Find its volume using the appropriate coordinate...
  45. S

    Evaluating Triple Integral: $\int\int\int_H(x^2+y^2) dV$

    How would I evaluate the triple integral \int\int\int_H(x^2+y^2) dV, where H is the region bounded x2 + y2 = 1, y = x, y = 0, z = 0, z = 2
  46. S

    How to evaluate a Triple Integral

    How would I evaluate the integral \int\int\int_{G} \sqrt{4x^2+9y^2} dV, where G is the elliptic cylinder 4x2+9y2 \leq 25, 0 \leq z \leq 6
  47. S

    Evaluating Triple Integral of G: xyz dV

    Given the triple integral \int\int\int_{G} xyz dV Where G is the region bounded by x=1, y=x, y=0, z=0, z=2. How do I evaluate it. Please help.
  48. P

    How to Evaluate a Triple Integral Bounded by a Cylinder and Planes?

    Homework Statement Evaluate the triple integral where E is the solid bounded by the cylinder y^2+ z^2 = 576 and the planes x = 0, y = 4 x and z =0 in the first octant. Homework Equations The Attempt at a Solution I figure that by solving for z I can get the bounds, so between 0 to...
  49. GRB 080319B

    What does the triple integral of a function represent graphically in 4D?

    If \int\int\int_{Q}dV = Volume_{Q}, and graphically, it represents the volume between all the boundaries of the respective variables in the iterated integral, what does \int\int\int_{Q}f(x,y,z)dV represent? Does this integral represent the (volume?) "above" (in 4D sense) the solid represented by...
  50. P

    Volume of Container - Triple Integral

    Homework Statement A container has a vertical height of 1m, a circular base with radius 1m and a circular top with radius 2m. Use a triple integral and spherical coordinates to evaluate the volume of the container. 2. The attempt at a solution If we set up the problem so that the centre of...
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