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. M

    Is My Solution for the Triple Integral Correct?

    Triple Integral Evaluation (quick and easy) Homework Statement \int_{0}^{1} \int_{x^2}^{1} \int_{0}^{3y} ({y+2x^2z})dz dy dx Homework Equations None. The Attempt at a Solution Here is what I got at the end (the LaTeX takes too long to code in here, plus its not showing up)...
  2. M

    Is My Triple Integral Calculation Correct or Is There an Error in the Book?

    Hi, my result of \int \int \int_{A} xyz dxdydz where A = \{(x,y,z); x^2+y^2+z^2 \leq 2, x \geq 0, y \geq 0, z \geq 0 \} is \frac {8}{48} , but book says \frac{1}{48} . Is the book right? Could you please verify? Thank you Michael
  3. T

    Bounds for triple integral

    Integrate the function f(x,y,z)=–6x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt((277/123))x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20. I...
  4. P

    Volume of Solid Inside Cylinder & Sphere

    Find the volume of the solid inside the cylinder x^2+y^2=2x and bounded by the sphere x^2+y^2+z^2=4 It appears that cyclindrical is out the question because there is no symmetry about the centre of the cylinder. So only spherical coords are applicable. Any clues?
  5. P

    Triple Integral: Find Region of Intersecting Cylinders

    I like to use cartesian coords Find the region to the intersecting cyclinders x^2+y^2<=a^2 and x^2+z^2<=a^2 What I have trouble finding is the domain of integration Currently I have a to -a for dx -srt(a^2-x^2) to srt(a^2-x^2) for dy -srt(x^2+y^2) to srt(x^2+y^2) for dz But this...
  6. T

    Triple Integral Evaluation with Spherical Coordinates

    Hi, I don't know whether this is the best place to ask, anyway, I would like to check my results in Maple and don't know how to evaluate something like this there: \iiint_{M} z^2\ dx\ dy\ dz\mbox{ , where M = [x,y,z] \in \mathbb{R}^3, x^2+y^2+z^2 \leq R^2, x^2 + y^2 + z^2 \leq 2Rz} There is...
  7. B

    How can I write a triple integral using different coordinate systems?

    Hello. This is my first of many posts at this forum. For fun recently I came across a triple integral that i would really like to know how to do. basically i have two equations x^2+y^2+z^2=1 x^2+y^2+(z-1)^2=2 if you plot these you will see that they intercept. i need to find the area...
  8. T

    Triple Integral Property: \bigtriangledown \times F

    "Suppose that a smooth vector field F(x,y,z) given on a region D has the property that on the bounding surface S it is perpenticular to the surface. Show that \int\int\int_D \bigtriangledown \times F dV = 0 in the sense that each component of \bigtriangledown \times F has integral 0 over D."...
  9. B

    Spherical coordinates triple integral

    Hi, Please can someone help me with this problem: find the triple integral over T( using spherical coordinate) T: 0<=x<=1 0<= y<=sqrt(1-x^2) sqrt(x^2+y^2)<= z <= sqrt(2-(X^2+y^2)) please help me just to find the boundaries of the integrals. I tried but I did not find the...
  10. W

    Triple integral - solid tetradhedon

    Evaluate the triple integral \int \int \int xy*DV where E is the solid tetrahedon with vertices (0,0,0), (4,0,0),(0,1,0),(0,0,7) first I'm going to find n: AB= <-4,1,0> AC= <-4,0,7> AB X AC = <7,28,4> = n so i get this equation: 7(x-4) + 28y + 4z = 0 => 7x+28y+4z = 28 so the...
  11. M

    How Do I Properly Convert a Triple Integral to Cylindrical Coordinates?

    Firstly, can someone please demonstrate the proper Latex code for the terminals on a multiple integral? Thanks! Anyway, as you can probably see, I'm calculating the volume enclosed by x^2+y^2+z^2 = 2 and z = x^2+y^2 using a change to cylindrical coordinates. V =...
  12. M

    How Do You Simplify the Triple Integral of |xyz| Over an Ellipsoid?

    I'm having trouble with evaluating [Triple Integral] |xyz| dx dy dz over the region (x/a)^2 + (y/b)^2 + (z/c)^2 <= 1 Do I need to use some sort of parametrisation for the region, and is there some way of dealing with the absolute value function without integrating over the eight...
  13. T

    How to Find Equivalent Triple Integrals?

    ok, so I've got this triple integral: \int_{0}^{2} \int_{0}^{y^3} \int_{0}^{y^2} f(x,y,z) dz\, dx\, dy\ what i want to do is get the other five integrals that are equivalen. I've got correctly 3 of them, but, for the life of me, cannot get dy dz dx and dy dx dz to work out. i've...
  14. V

    Setting up a triple integral in cylindrical coordinates?

    The problem says to find the volume of material cut from the solid sphere, r^2 + z^2 \le 9 by the cylinder, r = 3\sin\theta I don't know how to graph the first equation, but I can do the second in polar coordinates. How do I go about converting to use cylindrical coordinates?
  15. S

    How can we set up a triple integral to solve this problem?

    Triple Integral setup... \int \int \int_{G} 6x (z+y^3) dx dy dz G bounded by x = 0, \ x = y, \ z = y-y^2, \mbox{and} \ z=y^2 - y^3 x from 0 to1 y from 0 to x z from z=y-y^2 to y^2 - y^3 and the integration order becomes dz dy dx would this give the right answer? what aboiut this...
  16. V

    How Do You Set Up Limits for a Triple Integral Involving a Cylinder and Planes?

    Ok I hve a triple integral problem for find the area of the following: cylinder: x^2 + y^2 = 4 plane: z = 0 plane: x + z = 3 It is a cylinder cut at the xy-plane and by the last plane. It looks like a circular wedge standing straight up from the xy-plane. I just can't figure out...
  17. P

    Triple integral and center of mass

    A cone of height h and base radius r has density equal to distance from its base. Find it's center of mass. How do I write a function for the density? Is it p=h-z? And what are the limits of r if I want to do this in cylindrical coordinates? Thanks in adv.
  18. M

    Which Integration Technique Should I Use for This Triple Integral?

    \iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 \sin\theta\,dR\,d\theta\,d\phi with the integration over R between 0 and a the integration over between 0 and pi the integration over between 0 and 2pi Should I use integration by parts or should I take the R^2 sin(theta) under the square...
  19. T

    Triple Integral Help: Solving Equations and Finding Volume with Closed Curve C

    I have a group of problems that deals with the equations: f(x,y)= x^2+y^2 g(x,y)=20-(x-4)^2-(y+2)^2. I know that the surfaces z=f(x,y) and z=g(x,y) intersect in a closed curve, C, and the projection of C onto the xy-plane is a circle. However, I am having trouble finding its...
  20. cepheid

    Triple Integral in Spherical Coordinates

    I have a hemispherical surface of radius R with it's base centred on the origin. We are using the convention: r is the radius i.e. the magnitude of the position vector of a point: its distance from the origin. theta is the polar angle phi is the azimuthal angle I am asked to...
  21. T

    Another Triple Integral Question

    Hello. Here is the original question http:// My difficuly is understanding the limits of integration--- party due to how the solid in question is "sliced" by those planes. I know how to visualize the parabolic cylinder, but I need help on 1. limits on integration, and 2. Order of...
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