Triple integrals Definition and 81 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. E

    Rewriting triple integrals e.g. dz dy dx -> dx dy dz

    I'm having a tough time rewriting integrals from one form to another when the first integrand is not a function of two variables. As an example, when writing the integral to find the volume of a tetrahedron, I can easily write all 6 versions of the integral based on z = 1 - x - y or some...
  2. G

    Triple Integrals - Solve Boundaries of Integration

    Homework Statement Integrate \int_D z dxdydz where D is z\geq 0, z^2*\geq 2x^2+3y^2-1, x^2+y^2+z^2 \leq 3 Homework Equations Spherical coordinates? I'm stuck. I have problems finding the boundaries of integration. The Attempt at a Solution None. I'd be most grateful for help.
  3. J

    Triple Integrals. Need help I cant figure out why 2 integrals arent matching up

    Triple Integrals I am currently taking Calc 3 at college and I ran across a small problem with this one triple integral I am trying to work out (The problem is written as Example 1 in the book work document· So far I have had no luck in finding someone who could help me. If you can't help me...
  4. J

    Triple Integrals with Spherical Coordinates: Finding Limits

    Homework Statement I have this question about triple integrals and spherical coordinates http://img405.imageshack.us/img405/9343/81255254.th.jpg Homework Equations y = \rho sin \varphi sin \theta x = \rho sin \varphi cos \theta z = \rho cos \varphi \rho2 = z2 + y2 + x2...
  5. J

    Volume of a Solid using Triple Integrals

    Homework Statement Use a triple integral to find the volume of the solid enclosed by the cylinder z=y2 and the planes x=0, x=6, and z=16. Set up the integral in rectangular coordinates and work it out in any coordinates. Homework Equations The Attempt at a Solution I set up the triple...
  6. B

    How Do You Calculate the Average Temperature of a Solid Using Triple Integrals?

    Homework Statement A solid is definited by the inequalities 0\leqx\leq1, 0\leqy\leq1, and 0\leqz\leqx2+y2. The temperature of the solid is given by the function T=25-3z. Find the average temperature of the solid. The Attempt at a Solution I solved the integral, however I could not...
  7. A

    Calculating Volume using triple integrals

    Homework Statement please help me in determining the volume of the solid bounded by y^2 = 4x x^2 = 4y y = 3 x + y =3 z = x - y i need to use triple integrals Homework Equations v = v1 + V2 + V3 The Attempt at a Solution V1 = triple integral of dz dy dx
  8. J

    Geometric Series and Triple Integrals

    Homework Statement \int 1/(1-xyz)dxdydz = \sum1/n3 from n = 1 to infiniti dx 0 to 1 dy 0 to 1 dz 0 to 1 Homework Equations The Attempt at a Solution Not sure how to relate the two of them
  9. J

    What is the Region Bounded by a Unit Circle in the yz Plane and x = 0?

    Homework Statement evaluate the integral y2z2dV over W, which is the region bounded by x = 1 - y2 - z2 adn the plane x = 0 Homework Equations The Attempt at a Solution since x = 0, that makes y2 + z2 = 1, unit circle in the yz plane right? so would the answer be the area of...
  10. P

    TRIPLE INTEGRALS - How do I do this -I cannot draw at all

    I really can't draw at all, so usually i just imagine the figures in my head and then do it and then I usually imagine a slice perpedicular to some axis (eg x) take the double integral T (x) over the slice and then integrate that over x. The Region is bounded by the siz planes z=1...
  11. S

    How Do You Calculate the Volume of a Volcano Using Triple Integrals?

    Homework Statement A volcano fills the volume between the graphs z = 0 and z =1/(x^2+y^2)^24, and outside the cylinder x^2+y^2=1 Find the volume of this volcano. Homework Equations This is a triple integral to be evaluated in cylindrical coordinates. The Attempt at a Solution...
  12. W

    Total Mass: Calculating the Mass of a Lamina Using Triple Integrals

    Homework Statement A lamina occupies the part of the disk x^2 + y^2 ≤ 16 in the first quadrant and the density at each point is given by the function ρ(x,y) = 2(x^2+y^2) . What is the total mass? Where is the center of mass? (Once I solve total mass I can solve the center by myself.) The...
  13. W

    Find The Volume; Triple Integrals

    Find the volume of the solid enclosed by the paraboloids z = (x^2 + y^2 ) and z = 32 − ( x^2 + y^2) .To make this problem easier to look at I resorting to making it into cylindrical coordinates. {r, theta, z| 0< r< 1, 0<theta<2pi, r< z< 32-r} Every time I solve for this I end up getting 31pi...
  14. M

    Triple Integrals: Find Volume of Region Bounded by x+y, 10, 0, 0

    Homework Statement Find the volume of the region bounded by z=x+y, z=10, and the planes x=0, y=0 The Attempt at a Solution If I want to integrate with respect to z,y, then x; Then I think the limits of integration would be 0≤x≤z-y, so for x the be its largest, set y=0 and z to be...
  15. D

    Triple integrals in spherical & cylindrical coordinates

    Homework Statement Set up triple integrals for the volume of the sphere rho = 2 in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. Homework Equations Volume in cylindrical coordinates: Triple integral of dz r dr d(theta) over region D. Volume in spherical coordinates...
  16. H

    Limits for Rho in Triple Integral for Volume of Solid Bounded by Two Surfaces

    Homework Statement Find the volume of the solid bounded above by \rho=1+cos\varphi and below by \rho=1 Homework Equations The Attempt at a Solution I already solved it but was comparing my answer to my professor's solution, I was wondering why when he did the integration, his...
  17. C

    Does the Triple Integral Formula Apply to a Point-Mass Inside a Spherical Shell?

    Does -Gm\rho2\pi\left(R_2^2-R_1^2\right) make sense for the potential of a point-mass "m" inside a spherical shell of radii R_1< R_2 and density \rho? Now I've already found the potential outside of a homogeneous sphere of same density. I'm now asked to use these two results to find the...
  18. H

    Triple Integrals - Finding Mass

    The problem is the following: I need to find the mass, moments along the axis and the center of mass of the tetrahydron (centroid) with vertecies (-1,0,0) (1,0,0) (0,1,0) and (0,-1,0) and (0,0,2) basically it has a square base with an area of 4 and height 2 units. You are also given the...
  19. B

    Rewriting iterated triple integrals

    [SOLVED] Rewriting iterated triple integrals Homework Statement Rewrite this integral as an equivalent iterated integral in the five other orders. Homework Equations \int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z) dz dy dx The Attempt at a Solution Ok, so I have the shape...
  20. B

    Triple Integrals: Evaluating ∫∫∫6xydV

    Homework Statement Can someone see if I have set this up correctly? So I am to evaluate ∫∫∫6xydV. The region lies between z = 1+x+y and above the region in the xy plane bounded by the curves y = √x, y = 0, x = 1. So, would this be equal to ∫∫∫6xydzdydx, where z is evaluated from 0 to...
  21. U

    Triple Integral: Evaluating ∫∫∫E sqrt(x2+y2) dV

    evaluate \int \int \int _E \sqrt{x^2+y^2} dV, where E is the solid bounded by the circular parabola z=9-4(x^2+y^2) and the xy-plane so here's what I did, i tried to set this up in cylindrical coordinates. the radius: is when z=9-4(x^2+y^2) equals with the xy-plane so this means...
  22. denian

    What's the difference btw double and triple integrals

    i read both double and triple integrals can be used to find the volume. so, what's the difference?
  23. A

    Triple integrals in spherical coordinates

    i have a question concerning transforming triple integrals into spherical coordinates. the problem is, i do not know how to find the limits of phi. Can anyone help me? Thanks...
  24. O

    Why Is the Jacobian Determinant Used in Double and Triple Integrals?

    I understand double and triple integrals and all, but I'm just wondering why is dxdy=|J|dudv\ x=f(u,v)\ y=g(u,v) Where does that derive from? Why is it? (and also for triple integrals)
  25. S

    Triple Integrals: Solving G Bounded by x, 2-x, and y^2

    \int \int \int_{G} (xy + xz) dx dy dz G bounded by z=x, z= 2-x, and z = y^2 solving the first 2 i get x =1 equating y^2 = z =x and y^2 = 2-x so x can go from 1 to 2? not sure how to proceed for the y part, however.. please helppppp
  26. P

    Find Mass of Ball B w/ Triple Integrals & Cylindrical Coordinates

    Find the mass of a ball B given by "x^2+y^2+z^2≤a^2" if the density at any point is proportional to its distance from the z-axis using cylindrical coordinates So is the density equal to K*sqrt(x^2+y^2), or K*r? Using triple integral of f(rcosθ, rsinθ, z)*r*dz*dr*dθ) I got the following...
  27. W

    Finding The Volume Of Solid Using Triple Integrals II

    Hello, I am still unsure of my ability to evaluate the volume of a solid using triple integrals. Here is my question: Now I know that the intersection of the two paraboloids is 9 = x^2 + y^2. But I am unsure how to set up the triple integral. I was thinking of splitting the volume...
  28. W

    Finding The Volume Of Solid Using Triple Integrals

    Hello, I am having trouble setting up triple integrals to find a volume of a given solid. Here is one of the questions with which I am having trouble. Now I can see that the projection of the solid on the xy plane is the circle x^2 + y^2 = 9. And I think I can visualize the plane z = y +...
  29. W

    Triple Integrals Over General Regions

    Hello, First I will post my question: It has been quite a while since my last calculus course so I don't remember everything. Now here is MY question: How do I find the equation of the plane in which the region E lies below? I know from the solution manual that the E is the region that...
  30. Oxymoron

    Triple Integrals and the Circumference of a Unit Circle: An Exploration

    I was wondering if it was possible (just out of curiosity) to show that the circumference of a unit-circle is 2(pi) WITHOUT using the rule C = 2(pi)r? Is it possible to take a unit-sphere (a sphere with r=1) and triple integrate it. Something like... SSS sqrt(x^2 + y^2 + z^2)dxdydz = 2(pi)...
  31. D

    Solving triple integrals in a tetrahedron: How do I get started?

    Compute triple integral f(x,y)dV given f(x,y,z)=2x+3y. T is the tetrahedron bounded by the coordinate planes and first octant part of the plane with equation 2x + 3y + z = 6. how do i solve for this, can someone get me started halfway, please? Dx
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