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

    MHB Finding the Bounds for Evaluating a Triple Integral

    Evaluate the triple integral ∫∫∫E 5x dV, where E is bounded by the paraboloid x = 5y2 + 5z2 and the plane x = 5.
  2. P

    How to Determine the Centroid of a Solid Using Cylindrical Coordinates?

    Homework Statement Use cylindrical coordinates to find the centroid of the solid. The solid that is bounded above by the sphere x2 + y2 + z2 = 2 and below by z = x2 + y2 Homework Equations x = rcos(theta) y= rsin(theta)[/B]The Attempt at a Solution I am having trouble trying to find the...
  3. L

    Coulomb triple integral for a sphere

    Homework Statement Find the electric field of a sphere of radius R and charge Q outside sphere. Use only a Coulomb integral to do this. Homework Equations I know that I have to use a triple integral to find the E-field. I am just unsure of my whole setup really. The Attempt at a Solution...
  4. T

    Volume Inside a Sphere and Cone?

    Find the volume laying inside x^2 + y^2 + z^2 =2z and inside z^2 = x^2 + y^2. This is a problem my professor made, so I have no way of checking my answer. What I did first was completed the square for the sphere and got x^2 + y^2 + (z-1)^2 = 1, which is a sphere of radius one shifted above the...
  5. AdityaDev

    Acceleration due to gravity at the centre of a hemisphere

    Homework Statement find the acceleration due to gravity at the centre of a solid hemisphere. Homework Equations ##F=\frac{GMm}{r^2}## The Attempt at a Solution i decided to go for cylindrical coordinayes (which is way beyond my syllabus). I did some research though. let me take a point...
  6. H

    Triple integral in spherical coordinates

    Homework Statement Evaluate \int \int \int _R (x^2+y^2+z^2)dV where R is the cylinder 0\leq x^2+y^2\leq a^2, 0\leq z\leq h Homework Equations [/B] x = Rsin\phi cos\theta y = Rsin\phi sin\theta z = Rcos\phiThe Attempt at a Solution [/B] 2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta...
  7. S

    Triple integral for bounded regions

    Checking my steps and answer. Thanks in advance! Compute \int_0^3 \int_0^2 \int_1^3 xyz\ dz\ dy\ dx. \int_0^3 \int_0^2 \frac{xyz^2}{2} \Big|_1^3 = \frac{9xy}{2}-\frac{xy}{2} = \frac{8xy}{2} = 4xy \int_0^3 2xy^2 \Big|_0^2 \int_0^3 8x\ 4x^2 \Big|_0^3 = 36
  8. V

    Divergence Theorem Question (Gauss' Law?)

    If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...
  9. E

    MHB Triple Integral Bounded by Planes

    Hi All, Question: Consider the tetrahedron, T, bounded by planes x=2, y=0, z=0 and 3x-6y-2z=0. Determine the integral \iiintyDV which is the y coordinate of the centre of mass. I am getting a negative area which leads me to believe I'm doing something wrong. Working is attached. Help would...
  10. Calpalned

    Triple integral in cylindrical coordinates

    Homework Statement Evaluate ## \int \int \int_E {x}dV ## where E is enclosed by the planes ##z=0## and ##z=x+y+5## and by the cylinders ##x^2+y^2=4## and ##x^2+y^2=9##. Homework Equations ## \int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta ## How do I type limits in for...
  11. C

    Triple Integral for Divergence Theorem

    Homework Statement Find the flux of the field F(x) = <x,y,z> across the hemisphere x^2 + y^2 + z^2 = 4 above the plane z = 1, using both the Divergence Theorem and with flux integrals. (The plane is closing the surface) Homework Equations The Attempt at a Solution Obviously, the divergence...
  12. Amy Marie

    Transforming a Triple Integral: From Ellipsoid to Sphere

    Homework Statement Evaluate ∫∫∫[W] xz dV, where W is the domain bounded by the elliptic cylinder (x^2)/4 + (y^2)/9 = 1 and the sphere x^2 + y^2 + z^2 = 16 in the first octant x> or = 0, y> or = 0, z> or = 0. Homework Equations First, I tried to find the bounds for z: z = 0 (because z is...
  13. K

    Triple Integral of a cone bounded by a plane.

    Homework Statement find the volume using spherical coordinates of the region bounded above by z=9 and below by z=sqrt(x^2+y^2) in the first octant. Homework EquationsThe Attempt at a Solution I found this volume using cartesian and cylindrical coordinates, so I know the answer I am looking...
  14. K

    Triple Integral, Volume of an Egg

    Homework Statement I need to find the volume of an egg with a shape described by: z = 1/2(x2 + y2) and z = 6 - x2 - y2 I am also given that the egg is 6cm in length.Homework Equations I roughly graphed the two surfaces. The first being paraboloid that opens up from the origin, and the second...
  15. N

    What is the meaning of a triple integral?

    Okay so I just have a question on triple integrals. I understand how to use triple integrals to find volumes, but what I really don't understand is what I am really getting when I take the triple integral OF a function. I understand physical examples like taking the triple integral of a...
  16. 1

    Triple Integral using inequalties

    Homework Statement Evaluate \iiint z^2 \,dx\,dy\,dz over domain V, where V is the solid defined by 1 \leq x+y+3z \leq 20 \leq 2y-z \leq 3-1 \leq x+y \leq 1 Homework Equations The Attempt at a Solution I know how to do simple triple integrals, but all the variables in the inequalities are...
  17. J

    How Do You Set Up a Triple Integral in Spherical Coordinates for a Unit Ball?

    Homework Statement ##\iiint_W (x^2+y^2+z^2)^{5/2}## W is the ball ##x^2+y^2+z^2 \le 1## The Attempt at a Solution changing to spherical ##0 \le \theta \le 2\pi ; 0 \le \phi \le \pi ; 0 \le \rho \le 1## ##(x^2 + y^2 + z^2)^{5/2} \Rightarrow ((\rho \sin \phi \cos...
  18. X

    What is the purpose of the integrand in triple integration?

    Hi... So I've been self-teaching Calculus III and I'm currently having a hard time coping with the idea of triple integration. You know how the integrand is f(x,y,z)? isn't that the equation to represent a 4D sketch? because technically, f(x,y,z) is ANOTHER VARIABLE and therefore giving us a 4...
  19. F

    Volume of a tetrahedron by Triple Integral

    Homework Statement By using triple integral, find the volume of the tetrahedron bounded by the coordinate planes and the plane 2x+3y+2z=6.Homework Equations Volume= ∫vdv=∫∫∫dxdydz The Attempt at a Solution find intercepts of the plane on the axes, x-intercept=3 y-intercept=2...
  20. J

    Write a triple integral in spherical coordinates

    Homework Statement Write a triple integral in spherical coordinates that represents the volume of the part of the sphere X^2+Y^2+Z^2=16 that lies in the first octant(where x,y, and z are coordinates are all greater than or equal to zero) Homework Equations So i know this is in...
  21. J

    Triple Integral Problem in Cylindrical Coordinates

    Homework Statement Use cylindrical coordinates to find the volume of the solid that the cylinder r = 3cos/theta cuts out of the sphere of radius 3 centered at the origin. Homework Equations Why do we evaluate theta from 0 to pi instead of from 0 to 2pi? Don't we want to go all the...
  22. S

    Triple Integral Homework: Solve 0<=z<=y<=x^2<=1

    Homework Statement ∫∫∫z/(1+x^2)dxdydz with the range 0<=z<=y<=x^2<=1 Homework Equations The Attempt at a Solution I have tried with the limits 0<=z<=1; z<=y<=1 and sqrt(y)<=x<=1, but it doesn't get me the right answer. Can you please help me and maybe give me a step-by-step...
  23. J

    MHB Cylindrical Triple Integral Find the Volume?

    Let E be the solid inside cylinder y^2+z^2=1 and x^2+z^2=1, find the volume of e and the surface area of e
  24. A

    Changing the Order of Integration in this Triple Integral

    Homework Statement Evaluate ##\iiint_D (x^2+y^2)\mathrm{d}V##, where ##D## is the region bounded by the graphs of ##y=x^2##, ##z=4-y##, and ##z=0##. Homework Equations The Attempt at a Solution So after over at least an hour of thinking, I might have all 6 orders of integration...
  25. F

    Triple integral and Change of Variables

    Homework Statement The solid region W is bounded by the ellipsoid x^2/3 + y^2/5 + z^2/7 = 1. Find the triple integral ∫∫∫cos((35x^2 + 21y^2 + 15z^2)^(3/2))dV. Homework Equations Domain: x^2/3 + y^2/5 + z^2/7 = 1 Integral: ∫∫∫cos((35x^2 + 21y^2 + 15z^2)^(3/2))dV The Attempt at a...
  26. A

    Evaluating triple integral with spherical coordinates

    Homework Statement Evaluate the iterated integral ∫ (from 0 to 1) ∫ [from -sqrt(1-x^2) to sqrt(1-x^2) ] ∫ (from 0 to 2-x^2-y^2) the function given as √(x^2 + y^2) dz dy dx The Attempt at a Solution I changed the coordinates and I got the new limits as ∫(from 0 to pi) ∫(from...
  27. PsychonautQQ

    Evaluating a triple integral Spherical

    Homework Statement z(x^2+y^2+z^2)^(-3/2) where x^2+y^2+z^2 ≤ 4 and z ≥ 1 The Attempt at a Solution So spherically this comes down to cos∅sin∅dpdθd∅ p goes from 0 to 2, theta goes from 0 to 2pi, but I don't know how to figure out what ∅ goes from? I'm trying use trig identities but...
  28. PsychonautQQ

    Finding Volume Triple Integral

    Homework Statement Find the volume of area the area bounded by x^2+y^2=36, z=x and below the xy plane Homework Equations The Attempt at a Solution I did a triple integral dzdxdy, where dz is bound between x and 0 dx is bound between 0 and (36-y^2)^(1/2) and dy is bound between...
  29. L

    Need a hint for this triple integral

    Homework Statement Evaluate the triple integral of sin2z/(4-z) dydzdx where the limits of integration for outer limits (x) are from 0 to 2, the middle limits (z) are 0 to 4-(x^2), and the inner limits (y) are 0 to x.Homework Equations The Attempt at a Solution I'm not sure what the best...
  30. PsychonautQQ

    Triple Integral restricted to first octant.

    Homework Statement Evaluate the triple integral of function 14xz bounded between z=y^2 and z=8-2x^2-y^2 in the first octant. The Attempt at a Solution So the first octant would mean the bottom parameter on all my integral will be zero since (x,y,z)>0. Then I set the equations equal to...
  31. PsychonautQQ

    Triple Integral Example: Solving with the Order dydzdx and Correct Bounds

    Homework Statement Triple Integral (x^6e^y)dV bounded by z=1-y^2 z=0 x=-1 x=1 The Attempt at a Solution So I chose to try to integrate this in the order dydzdx My bounds for the dy integral were from zero to (1-z)^(1/2) my bounds for the dz integral were from 0 to 1 and my bounds...
  32. P

    Calculating Volume of Solid Using Triple Integral

    1. Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder x^2 + z^2 = 4 and the planes y = -1 and y + z = 4 This looked like a cylindrical coordinate system to me, except for the fact that it is not cylindrical around the z-axis but the y-axis. I...
  33. Petrus

    MHB Triple integral, spherical coordinates

    Hello MHB, So when I change to space polar I Dont understand how facit got \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2} Regards, |\pi\rangle \int\int\int_D(x^2y^2z)dxdydz where D is D={(x,y,z);0\leq z \leq \sqrt{x^2+y^2}, x^2+y^2+z^2 \leq 1}
  34. J

    Triple integral problem: cylindrical coordinates

    Homework Statement I have a graph 1/x^2=y^2+z^2 where z=rsin(θ) and y=rcos(θ) where 0≤r≤1 and 0≤θ≤2∏ on the zy-plane The end result is attached (sorry, I'm not aware of how to use Latex :[ ) I can kind of understand how they determined the first bounds for the integral: the lowest x...
  35. G

    Triple integral, limits of integration

    Pretty general question. Integrate f(x,y,z) dxdydz over the area defined by: x^{2} + y^{2} + z^{2} \leq 4 x \leq 0 y \leq 0 z \leq 0 It is immidiately apparent that it is 1/8 of a sphere with r=2. So from that geometrical intuition we can do a variable substitution to spherical...
  36. N

    Extremly long Definite triple integral

    Hi I am currently working on a project, and I need to calculate the definite triple integral of 1/|x+y+z|. i.e: int int int (1/sqrt((x-x')^2+(y-y')^2+(z-z')^2)) dx dy dz. I have solved the integral, and it has 48 terms if the limits are inserted (6 terms with 3 sets of upper and lower...
  37. K

    Triple Integral for earth potensial

    Hi! I am trying to solve this triple integral for computing the potential: G*ρ*∫∫∫-atan((x*y)/(z*sqrt(x^2+y^2+z^2)))dxdydz with the limits x = 0 to 1000 z = 0 to 1000 and z = 0 to 1. The G is tha gravitational constant and ρ is the density of rock/earth. I have tried to use multiple...
  38. R

    Center of Mass (Triple Integral)

    Homework Statement T is the solid bounded by the cylinder y^2+z^2=4 and the planes x=0 and x=3. The mass density at a point P of T is directly proportional to the distance between P and the yz-plane. Find the center of mass of the solid T. Homework Equations y^2+z^2=4 x=0 x=3 The...
  39. P

    Triple Integral over Region: Finding Volume with f(x,y,z) = z

    Homework Statement Let W be the region bounded by y + z = 2, 2x = y, x = 0, and z = 0. Express and evaluate the triple integral of f (x, y, z) = z by projecting W onto the: (a) xy-plane (b) yz-plane (c) xz-plane.Homework Equations The function f (x, y, z) = z and the boundary W: {y + z = 2, 2x...
  40. C

    Triple integral problem - domain is that part of a cube btween 2 plane

    Homework Statement Evaluate the triple integral for the function \int\int\int y dV over that part of the cube 0 \leq x,y,z \leq 1 lying above the plane y +z = 1 and below the plane x+y+z = 2 Homework Equations The Attempt at a Solution This is the first attempt at a triple...
  41. tsuwal

    Triple integral in eletromagnetism: prove force doesnt depend radius.

    Homework Statement This problem may be dull, I know, but maybe there is a hidden math trick that i don't know of. This picture sums up the problem. So, you should prove by simplifing the integral that F^e, the eletric force applied between two spheres, onde with a charge q_1 and the...
  42. E

    Definite Triple Integral to Series

    Homework Statement Does the triple integral \int^{1}_{0}\int^{1}_{0}\int^{1}_{0}\frac{1}{1+x^2 y^2 z^2} = \sum^{∞}_{n=0}\frac{1}{(2n+1)^3} Homework Equations The Attempt at a Solution I've not a single clue on what to do with this problem. I figured maybe I could find a decent...
  43. D

    MHB Solving a Triple Integral with $\rho$ Constant

    Anyone have any smooth ideas for this triple integral? $$ -\rho Gm\iiint\frac{r'^2\sin\theta}{\sqrt{R^2 + r'^2 - 2Rr'\cos\theta}}d\theta d\rho dr' $$ where $0<r'<a$, $0<\theta<\pi$, and $0<\rho<2\pi$. The $\rho$ out front is constant.
  44. T

    Is the Projection of a Triple Integral's Base Always a Rectangle?

    Homework Statement Here is the question along with the solution and sketch. I think the sketch is wrong because the projection in the xy plane shows a rectangular box. I don't think it is a rectangular box because you can solve for an equation relating x and y. You know that y = 1-z...
  45. L

    Triple integral in cylindrical coordinates

    Homework Statement Find the volume of the solid that lies between z=x2+y2 and x2+y2+z2=2 Homework Equations z=r2 z=√(2-r2) The Attempt at a Solution So changing this into cylindrical coordinates, I get z goes from r2 to √(2-r2) r goes from 0 to √2 theta goes from 0...
  46. D

    Triple integral depending on a parameter

    Homework Statement Find \iiint (x^{2n} + y^{2n} + z^{2n})\,dV where the integral is taken over the region of 3D space where x^{2} + y^{2} + z^{2} \leq 1 Homework Equations The Attempt at a Solution I tried doing this in Cartesian coordinates, but the limits of integration got...
  47. B

    Converting a triple integral from spherical to cartesian, cylindrical coordinates

    Homework Statement Consider the interated integral I=∫∫∫ρ^3 sin^2(∅) dρ d∅ dθ -the bounds of the first integral (from left to right) are from 0 to pi -the bounds of the second integral are from 0 to pi/2 -the bounds of the third integral are from 1 to 3 a)express I as an interated...
  48. N

    So the integral is a triple integral with the limits described above.

    Homework Statement Find the volume of the object, defined by these inequalities(?): x^2+y^2+z^2≤4, (x-1)^2+y^2≥1, (x+1)^2+y^2≥1Homework Equations The Attempt at a Solution First we draw the object, and realize that it's a sphere with 2 circles in it with radius 1 at (-1,0) and (1,0). Our...
  49. M

    Setting up a Triple Integral to Find Volume Enclosed by Two Functions

    Homework Statement Find the volume of the region of space enclosed between the functions: 1=-z+2x+2y and 100=z2+y2+x2. The Attempt at a Solution I am not sure how to set this problem up. I think it is a triple integral, since there is a z-component. I graphed the equation in an program and...
  50. STEMucator

    Quick question about a triple integral

    Homework Statement Let G be the region bounded by z=x2, z=y2 and z=3. Evaluate : \iiint\limits_G |xy| dV Homework Equations The Attempt at a Solution So fixing x and y didn't really give me any useful information. When I fixed z though, I got x=±z and y=±z which forms a square in the...
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