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

    Setting Up Triple Integral for Region Bounded by x=y^2, z=0, x+z=1

    Homework Statement Set up the triple integral for the region bounded by: x=y^2, z=0, x+z=1Homework Equations The Attempt at a Solution y= ±sqrt(x); z=0 & z=x+1 I'm just lost on how to find the x integral. I know the dz integral goes from z=0 to z=x+1, and I know the dy integral goes from...
  2. P

    Wattage delivered using triple integral help

    Useful equation. Avg. Power p(t)=(V^2(t))/R My attempt at instantaneous power was p(t,V,R)= ∫(0->1 for t ∫0->5 for V and ∫0->.1 for R V^2(t)/RdvdRdt Integrating I go the triple integral of V^3t^2/6R^2 Substituting my values in gave a wattage of 1,250 watts/m^2 at t=1 second, v=5...
  3. A

    How Do You Solve the Third Integral in a Triple Integral Problem?

    Homework Statement the limits of integration are s[0,R] p[0,2π] z[0,L] respectively kp∫∫∫s/sqrt(s^2+(d-z)^2)dsdpdz Homework Equations The Attempt at a Solution You can double check my work but I'm pretty sure I got the first to integrations fine (with respect to s and p)...
  4. T

    Evaluating Triple Integral in 1st Octant: Q & 3x+2y+z=6

    Homework Statement Evaluate \underbrace{\int\int\int}_{Q}(1-x) dzdydx Where Q is the solid that lies in the first octant and below the plane: 3x + 2y + z = 6 The Attempt at a Solution I guess my main problem is finding the integral limits. For dz, I arranged the equation of the...
  5. D

    Triple integral w/ spherical subsitution

    Homework Statement f(x) is a differentiable function let F(t)= \int\int\int_{x^2+y^2+z^2\leq t^2} f(x^2+y^2+z^2) dx dy dz compute F^{'}(t) Homework Equations x=p sin \phi cos\theta y= p sin \phi sin\theta z= p cos \phi spherical bounds 0<p<t 0<\phi<\Pi 0<\theta < 2\Pi p^2...
  6. Telemachus

    Volume Calculation of Region w/ Triple Integral

    Hi, I have to compute de volume for the region: \{(x,y,z):x^2+y^2+z^2\leq{16};z\geq{1} \} I've tried to do by two different parametrizations, in spherics and cylindrical coordinates For cylindrical coordinates I've made: 3 \displaystyle\int_{0}^{\sqrt[ ]{15}}\int_{0}^{2\pi}\int_{1}^{\sqrt[...
  7. N

    Triple integral in spherical form

    consider this following triple integral 1/(x^2+y^2+z^2)dxdydz bounded above by sphere z=(9-x^2-y^2)^1/2 and below by the cone z=(x^2+y^2)^1/2 what i have done: z=Pcospi P^2=x^2+y^2+z^2 9=x^2+y^2+z^2 P=0 to 3 pi=0 to pi/4 theta=0 to 2pi is this the correct range?
  8. I

    Triple Integral Evaluation for Bounded Region with Polynomial Boundaries

    Homework Statement Evaluate triple integral of 3xy over the bounded region: y = x^{2} x = y^{2} z = 6x + y The Attempt at a Solution Bounds on integral would be: 0 \leq x \leq 1 x^{2} \leq y \leq \sqrt{x} 0 \leq z \leq 6x + y Correct?
  9. I

    Triple Integral: Volume of a Solid

    Homework Statement Been awhile since I've done them and my memory/reasoning isn't so great apparently... Use the triple integral to find the volume of the given solid: The solid enclosed by the cylinder x^{2} + y^{2} = 9 and the planes y + z = 16 and z = 1. 2. The attempt at a solution...
  10. N

    Triple integral (spherical coordinate)

    find the volume of the solid D that lies above the cone z = (x^2 + y^2)^1/2 and below the sphere z = (x^2 + y^2 + z^2) i've done the integration until i need to substitute cos phi = u.. however.. i don't know to change the range.. http://imageshack.us/photo/my-images/839/spherical.jpg/"
  11. N

    Volume of Paraboloid-Bounded Solid in Cylindrical Coordinates?

    Find the volume of the solid that lies under the paraboloid z = x^2 + y^2, above xy plane, and inside the cylinder x^2 + y^2 = 2y First, i try to find the range.. i transfer it to cylindrical coordinates.. sqrt(y^2)=<z<=r^2 i don't know how to find r i know that phi is from 0 to 2pi...
  12. J

    Centroid of a 3D Region using Triple Integral

    Homework Statement Compute the centroid of the region defined by x^{2} + y^{2} + z^{2} \leq k^{2} and x \geq 0 with \delta(x,y,z) = 1. Homework Equations \overline{x}=\frac{1}{m}\int\int\int x \delta(x,y,z) dV \overline{y}=\frac{1}{m}\int\int\int y \delta(x,y,z) dV...
  13. J

    Moment of Inertia using Triple Integral

    Homework Statement Compute the moment of inertia around the z-axis of the solid unit box [0,1]x[0,1]x[0,1] with density given by \delta=x^{2}+y^{2}+z^{2}. Homework Equations I=\int\int\intr^{2} \delta dV The Attempt at a Solution I know that the distance r^{2} from the z-axis would...
  14. B

    What is the Correct Result for This Triple Integral Problem?

    Homework Statement I=(triple intgral)(x2+y2)dxdydz. D: z=2 x2+y2=2z z>=0 Homework Equations The Attempt at a Solution I used cylindrical coordinates to solve this. But I came across a problem. When I fix z between 0 and 2, and r between 0 and sqrt(2z) I get 16pi/3 {0<z<2...
  15. P

    Triple Integral moment of inertia

    set up a triple integral for the moment of inertia Iz for the region inside the sphere x^2+y^2+z^2=4a^2 and inside the cylinder x^2+y^2-2ax=0 so I draw my picture and convert to cylindrical coord. and i get an integral from 0 to sqrt(4a^2-r^2) an integral from 0 to 2acostheta and an...
  16. I

    How do you define the limits for this triple integral?

    I had this on a test today. First everything seemed easy, but then I got stuck. So the body over which the integral is to be taken is defined by: 1 <= x2 + y2 + z2 <= 4 and z >= sqrt(x2 + y2) Right now as I'm typing this I just thought that, why not plug z from the second eq. into the...
  17. H

    Converting Polar Triple Integral to Spherical One

    Homework Statement Evaluate the following triple integral by switching it to spherical coordinates? The integrand is r dzdrdθ The limits for the inner integral are 0 to r The limits for the middle integral are 0 to 3 The limits for the outer integral are 0 to 2π Homework Equations...
  18. L

    Suitable change of variables for this triple integral?

    Homework Statement [PLAIN]http://img542.imageshack.us/img542/5600/unledsn.png Homework Equations The Attempt at a Solution The first part is fine, just struggling to find a change of variables that'll help, tried spherical due to the x^2+y^2+z^2, didn't help enormously Thanks! (from sheet...
  19. T

    Converting to Cylindrical Coordinates for Triple Integration

    I don't want the answer, just a little help getting there. The question asks to integrate this: Triple integral I'm thinking to convert it to cylindrical but I have no idea how to convert the bounds. I can convert the actual expression z/sqrt(x^2+y^2) into cylindrical no problem. If I had...
  20. S

    Solving for Volume: Triple Integral of Wedge in Cylinder

    Homework Statement Why does 2*Integral (r dz dr dt) for 0<z<rcos(t), 0<r<a, 0<t<pi/2 equal (2a^3)/3, when Integral (r dz dr dt) for 0<z<rcos(t), 0<r<a, 0<t<pi equal 0? All you are doing is using the fact that rcos(t) is an even function to make the limits easier, right? Homework Equations...
  21. W

    Finding volume using triple integral

    1. Homework Statement I need to find the volume of a solid formed by the following equations: x^2+y^2 > 1 x^2+z^2 = 1 z^2 + y^2 =1 3. The Attempt at a Solution I know that it is a triple integral and the integrand is 1. I also know that I need to use dzrdrd\theta.
  22. B

    Triple Integral Help: Find Mass in Cylindrical Coords

    Find the mass of the region (in cylindrical coordinates)r^3<=z<=1 , where the density function is r(r;q; z) = 9z. This is what I got so far 0 <= r<= 1; 0 <= q <= 2p: hence need to compute the integral Z from 0 to 2pi Z from 0 to 1 Z r^3 to 1 9zdzrdrdq: We thus obtain 9p Z 0 to...
  23. W

    Finding volume using triple integral

    Homework Statement I need to find the volume of a solid formed by the following equations: x^2+y^2 > 1 x^2+z^2 = 1 x^2 + y^2 =1The Attempt at a Solution I know that it is a triple integral and the integrand is 1. I also know that I need to use dzrdrd\theta. I believe that you need two...
  24. P

    What are the bounds for evaluating a triple integral in the first octant?

    Homework Statement evaluate triple integral of z.dV where the solid E is bounded by the cylinder y2+z2=9 and the planes x=0 and y=3x and z=0 in the first octant Homework Equations for cylindrical polar co-ords, x=rcos\theta, y=rsin\theta and z=z The Attempt at a Solution im just...
  25. L

    Finding the bounds of a triple integral in cylindrical coordinates?

    Homework Statement I took a picture of the problem so it would be easier to understand. All I need to know is what the bounds are. Homework Equations In cylindrical: x=rcos(theta) y=rsin(theta) z=z The Attempt at a Solution I don't know why we should change this to...
  26. S

    Triple Integral of Tetrahedron

    Homework Statement Evaluate the triple integral \int\int\int^{}_{E} xy dV where E is the tetrahedron (0,0,0),(3,0,0),(0,5,0),(0,0,6). Is there a simple way to simplify the integration? Homework Equations The Attempt at a Solution \frac{z}{6} + \frac{y}{5} + \frac{x}{3} = 1 z =...
  27. S

    Triple integral from cartesian to spherical coordinates

    Homework Statement evaluate the following triple integral in spherical coordinates:: INT(=B) = (x^2+y^2+z^2)^2 dz dy dx where the limits are: z = 0 to z = sqrt(1-x^2-y^2) y = 0 to z = sqrt(1-x^2) x = 0 to x = 1 Homework Equations The only thing I know for sure is how to set...
  28. K

    Is It Possible to Simplify This Tricky Triple Integral?

    \int^{1}_{0}\int^{x/2}_{0}\frac{y}{(2y-1)\sqrt{1+y^2}}dydx Most of my attempts at this problem fail pretty quickly. Not even my calculator knows what to do with this one.
  29. S

    Volume between a sphere and cone using triple integral

    Homework Statement Evaluate the volume inside the sphere a^2 = x^2+y^2+z^2 and the cone z=sqrt(x^2+y^2) using triple integrals.Homework Equations a^2 = x^2+y^2+z^2 z=sqrt(x^2+y^2) The solution is (2/3)*pi*a^3(1-1/sqrt(2)) The Attempt at a Solution I first got the radius of the circle of...
  30. B

    Triple Integral Limits Help. Cylindrical Coordinates

    Homework Statement Find the volume of the solid bounded by the paraboloids z=x^2+y^2 and z=36-x^2-y^2. Answer is: 324\pi \\ Homework Equations r^2=x^2+y^2 x=rcos0 y=rcos0 The Attempt at a Solution 36-x^2+y^2=x^2+y^2\\ 36=2x^2+2y^2 18=x^2+y^2 r^2=18 V=\int_{0}^{2\pi} \int_0^{3\sqrt{2}}...
  31. B

    Finding Volume Using Triple Integrals with Non-Standard Limits

    Homework Statement Find the volume of the region in the 1st octant bounded above by the surface z=4-x^2-y and below by the plane z=3. Answer = 4/15Homework Equations V = \int\int\int dVThe Attempt at a Solution I'm having trouble determining the upper and lower z limits. I partly don't...
  32. D

    Why is the last term on the RHS missing in my evaluated triple integral?

    Hi, I'm having a problem in evaluating a triple integral for a deformable control volume equation: where v is defined as: When I evaluate the triple integral in Maple and by hand I get: The correct answer is: Can someone please explain where the last term on the RHS comes...
  33. M

    How to Correctly Adjust Integration Bounds for a Complex Triple Integral?

    I= [∫(0 to 1) ∫(0 to 2z) ∫(z to 1) dx dy dz] + [ ∫(0 to 1) ∫(2z to 1+z) ∫(y-z to 1) dx dy dz] 1) evaluate 2) use order dy dx dz, along with the new bounds my attempt for 1) got me an answer of -7/6 for part 2) I'm having trouble getting the correct bounds. the bounds from my attempt...
  34. M

    Finding three orders of integration for a triple integral over unusual region

    Homework Statement 2. The attempt at a solution It's not hard to find two orders of integration. (1) Integrate first with respect to x_3, then with respect to x_2, and then with respect to x_1, by dividing D into two regions: D = \{x \in R^3 \mid -1 \leq x_1 < 0, -\sqrt{1-x_1^2}...
  35. G

    Calculate a Triple Integral using the cylindrical coordinate system

    I don't understand what's wrong with it:
  36. A

    Triple Integral, Volume of a solid

    Homework Statement Well, first of all, I'm not english spoken, so sorry for the mistakes. I was trying to calculate the integral below: \int \int \int_{V} (xy+z) dxdydz where V is a region in R^{3} bounded by the sphere x^2+y^2+z^2<=9 the cone z^2<=x^2+y^2 and the plane...
  37. S

    Triple Integral, Spherical Coordinates

    Homework Statement Calculate: integral B [ 1/sqrt(x^2+y^2+(z-a)^2) ] dx dy dz , when B is the sphere of radius R around (0,0,0), a>R. Homework Equations The Attempt at a Solution I tried spherical coordinates for the integrand: x=rsin(p)cos(t) y=rsin(p)sin(t) z=rcos(p)+a The problem is when I...
  38. B

    Triple Integral in Spherical Coordinates

    Evaluate the integral by changing to spherical coordinates. Not sure how to go about figuring out the limits of integration when changing to spherical coordinates.
  39. M

    Triple Integral: Evaluating Limits

    Homework Statement Evaluate the triple integral. ∫∫∫xyz dV, where T is the sold tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), and (1,0,1)The Attempt at a Solution I'm having trouble finding the bounds. So far I'm integrating it in order of dzdydx with my x bounds as 0-1, my y bounds as...
  40. jegues

    Triple Integral over the volume bounded by

    Homework Statement Evaluate the triple integral of the function f(x,y,z) = x over the volume bounded by the surfaces 2x + 3y + z =6,x=0,y=0,z=0. Homework Equations The Attempt at a Solution See figure attached for my attempt. I sketched the volume bounded by the surfaces...
  41. P

    What are the bounds for this strange triple integral over the region E?

    Homework Statement {\int}{\int}{\int}ydV over the region E, where E is bounded by x=0, y=0, z=0, and 2x+2y+z=4 Homework Equations n/a The Attempt at a Solution Assuming that x and y must both be positive, which the boundary conditions seem to require, then the most either one can...
  42. Telemachus

    Triple integral doubt, spherical coordinates

    Hi there. I have some doubts about this exercise. It asks me to use the appropriated coordinates to find the volume of the region indicated below. The region is determined by the first octant under the sphere x^2+ y^2+ z^2=16 and inside the cylinder x^2 +y^2=4x The first thing I did was to...
  43. T

    How to Calculate the Volume of a Solid Bounded by a Sphere and a Cone?

    Homework Statement Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 =4, above the xy-plane, and below the cone z=sqrt(x^2 + y^2). The Attempt at a Solution Use Cylindrical Coordinates. Note that r ≤ z ≤ √(4 - r^2). These sphere and cone intersect when x^2 + y^2 +...
  44. S

    Finding Volume Using Triple Integrals: A Brief Guide

    Homework Statement i have to find the volume between the function z=4-x^2-y^2 and the x/y plane The Attempt at a Solution I think I should be fine with the limits of integration but am not 100% confident what I am integrating. is it 4-x^2-y^2-z?? or 4-x^2-y^2?
  45. B

    Triple integral with an image given

    Homework Statement Integrate the function f(xyz)=−4x+6y over the solid given by the figure below, if P = (5,1,0) and Q = (-5,1,5). where P=(5,1,0) and Q=(-5,1,5) Homework Equations r²=x²+y² tan theta=y/x z=z y=rsintheta x=rcostheta The Attempt at a Solution So I treated this...
  46. B

    Triple Integral problem (maybe using spherical coordiantes)

    Homework Statement Integrate the function f(xyz)=−4x+6y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=x*sqrt(22/3) and contained in a sphere centered at the origin with radius 13 and a cone opening upwards from the origin with top...
  47. M

    Triple Integral (Calc III-Multivariable Calc)

    Homework Statement Integrate the function f(x,y,z)=3x+8y 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(17/47))*x and contained in a sphere centered at the origin with radius 10 and a cone opening upwards from the origin...
  48. L

    How Do You Calculate the Triple Integral of (x+5y) in a Bounded Region?

    Homework Statement Evaluate the triple integral of (x+5y)dV where E is bounded by the parabolic cylinder y=3x^2 and the planes z=9x, y=18x and z=0. Homework Equations The Attempt at a Solution My solution is this... 27*6^5 /5 - 162*6^4 /4 + (45/2)*(9*6^6 /6 - 324*6^4 /4) =...
  49. L

    Finding bounds of triple integral

    Homework Statement Evaluate the triple integral of the function (x+5y)dV Where E is bounded by a parabolic cylinder and the planes z=9x z=0 y=18x and y=3x^2 I just wanted to knw if my bounds are correct. Here they are for dz: 0 to 9x dy: 18x to 3x^2 dx: 0 to 6
  50. L

    Triple Integral of xy over a Solid Tetrahedron | Homework Statement

    Homework Statement Evaluate the triple integral of xy*dxdydz where E is the solid tetrahedon with vertices (0,0,0) (10,0,0) (0,8,0) (0,0,5). The Attempt at a Solution Im trying to integrate dx and dy with bounds from 0 to the line that describes them with respect to the z axis, so for...
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