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. mncyapntsi

    Confused about polar integrals and setting up bounds

    So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds. Any help would be greatly appreciated! Thanks!!
  2. A

    Why this triple integral is not null?

    Greetings here is my integral Compute the volume of the solid and here is the solution (that I don't agree with) So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric...
  3. A

    Problem with a triple integral in cylindrical coordinates

    Good day here is the solution J just don't understand why the solution r=√2 has been omitted?? many thanks in advance best regards!
  4. D

    Is the Triple Integral in Cylindrical Coordinates Correctly Solved?

    I am trying to solve it using cylindrical coordinates, but I am not sure whether the my description of region E is correct, whether is the value of r is 2 to 4, or have to evaluate the volume 2 times ( r from 0 to 4 minus r from 0 to 2), and whether is okay to take z from r^2/2 to 8
  5. O

    I Am I using the right limits on this triple integral?

    Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} Consider the expression: \begin{align} \int_{q'}^q \int_{b'}^b g(s)\ \int_{s-t}^{s+t} f(r)\ dr\ ds\ dt\ \end{align} We...
  6. D

    Represent a 3d region and compute this triple integral

    Let ## E=\left\{ (x,y,z) \in R^3 : 1 \leq x^2+y^2+z^2 \leq 4, 3x^2+3y^2-z^2\leq 0, z\geq0 \right\} ## - Represent the region E in 3-dimensions -represent the section of e in (x,z) plane -compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz## the domain is a sphere of radius 2 with an inner...
  7. karush

    MHB 15.1.34 Evaluate triple integral

    15.1.34 Evaluate $\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}} \sin{y} \, dz \, dx \, dy$ integrat dz $\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy $ integrat dx $\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$...
  8. WMDhamnekar

    MHB Computing Triple Integral in 'R'

    I would like to compute the triple integral of a function of three variables $f(x,y,z)$in R. I am using the package Cubature, Base, SimplicialCubature and the function adaptIntegrate(), Integrate and adaptIntegrateSimplex(). The integrand is equal to 1 only in certain domain(x<y<z, 0 otherwise)...
  9. aligator11

    Multivariable Triple Integral - Calculus Physics/Math Problem

    Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...
  10. TheMercury79

    Why does Mathematica return twice this value?

    The integral is$$\int_0^4dz\iint xyz~dxdy$$Constricted to the quarter circular disk ##x^2+y^2=4## in the first quadrant. First I switched to polar coordinates and integrated the double integral by first writing it as:$$\int_0^4z~dz \int_0^\frac{\pi}2\int_0^2...
  11. JD_PM

    Changing the order of a triple integral

    Homework Statement $$\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) dxdzdy$$ Change the order of the integral to $$\iiint f(x,y,z) \, \mathrm{dydzdx}$$What I have done It is just about: From ##x=0## to ##x=\sqrt{4y+23}## From ##z=0## to ##z=4-y## From ##y=\frac{x^2-23}{4}## to...
  12. M

    B Triple integral in spherical coordinates.

    While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...
  13. E

    I Closed-form solution for a triple integral

    Hello all, I need to evaluate the following 3-dimensional integral in closed-form (if possible) \int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min(x_2,\,y_1(z-\frac{x_2}{y_2}))\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1 where ##z## is real positive number, and...
  14. B

    Use triple integral to find center of mass

    Homework Statement Find the centre of mass of a uniform hemispherical shell of inner radius a and outer radius b. Homework Equations ##r_{CoM} = \sum \frac{m\vec{r}}{m}## The Attempt at a Solution Using ##x(r,\theta,\phi)## for coordinates...
  15. F

    Cylindrical Coordinates Triple Integral -- stuck in one place

    Homework Statement Use cylindrical coordinates to evaluate triple integral E (sqrt(x^2+y^2)dv where E is the solid that lies within the cylinder x^2+y^2 = 9, above the plane z=0, and below the plane z=5-y Homework EquationsThe Attempt at a Solution So i just need to know how to get the bounds...
  16. karush

    MHB 213.15.4.17 triple integral of bounded by cone and sphere

    $\textsf{Find the volume of the given solid region bounded by the cone}$ $$\displaystyle z=\sqrt{x^2+y^2}$$ $\textsf{and bounded above by the sphere}$ $$\displaystyle x^2+y^2+z^2=128$$ $\textsf{ using triple integrals}$ \begin{align*}\displaystyle V&=\iiint\limits_{R}p(x,y,z) \, dV...
  17. karush

    MHB What is the value of the triple integral for the given limits and function?

    \begin{align}\displaystyle v_{\tiny{s6.15.6.3}}&=\displaystyle \int_{0}^{1}\int_{0}^{z}\int_{0}^{x+z} 6xz \quad \, dy \, dx\, dz \end{align} $\text{ok i kinda got ? with $x+z$ to do the first step?}\\$ $\text{didn't see an example}$
  18. karush

    MHB How do you evaluate the spherical coordinate integral at 244.15.7.24?

    $\tiny{244 .15.7.24}$ $\textsf{Evaluate the spherical coordinate integral}\\ \begin{align}\displaystyle DV_{24}&=\int_{0}^{3\pi/4} \int_{0}^{\pi} \int_{0}^{1} \, 5\rho^3 \sin^3 \phi \, d\rho \, d\phi \, d\theta \\ &=\int_{4}^{3\pi/4}...
  19. karush

    MHB 244.T.15.5.11 Evaluate the triple integral

    $tiny{244.T.15.5.11}$ $\textsf{Evaluate the triple integral}\\$ \begin{align*}\displaystyle I_{\tiny{11}}&=\int_{0}^{\pi/6}\int_{0}^{1}\int_{-2}^{3} y\sin{z} \, d\textbf{x} \, d\textbf{y} \, d\textbf{z}\\ &=\int_{0}^{\pi/6}\int_{0}^{1}...
  20. karush

    MHB What is the value of the triple integral 15.4.08?

    \begin{align*}\displaystyle I_{15.5.8}&=\int_{0}^{\sqrt{2}} \int_{0}^{3y} \int_{x^2+3y^2}^{8-x^2-y^2} dz \ dy \ dx \\ &=\int_{0}^{\sqrt{2}} \int_{0}^{3y} \Biggr|z\Biggr|_{x^2+3y^2}^{8-x^2-y^2}\\ &=\int_{0}^{\sqrt{2}} \int_{0}^{3y} 8-2x^2-4y^2 \ dy \ dx \\ &=\int_{0}^{\sqrt{2}}\Biggr|8y-2x^2...
  21. S

    Triple Integral of y^2z^2 over a Paraboloid: Polar Coordinates Method

    Homework Statement Evaluate the triple integral y^2z^2dv. Where E is bounded by the paraboloid x=1-y^2-z^2 and the place x=0. Homework Equations x=r^2cos(theta) y=r^2sin(theta) The Attempt at a Solution I understand how to find these three limits, -1 to 1 , -sqrt(1-y^2) to sqrt(1-y^2) , 0 to...
  22. M

    MHB Calculating a Triple Integral in a Bounded Region

    Hey! :o Let $D$ be the space $\{x,y,z)\mid z\geq 0, x^2+y^2\leq 1, x^2+y^2+z^2\leq 4\}$. I want to calculate the integral $\iiint_D x^2\,dx\,dy\,dz$. I have done the following: We have that $x^2+y^2+z^2\leq 4\Rightarrow z^2\leq 4-x^2-y^2 \Rightarrow -\sqrt{4-x^2-y^2}\leq z\leq...
  23. Draconifors

    Finding center of mass of solid

    Homework Statement A solid B occupies the region of space above ##z=0## and between the spheres ##x^2 + y^2 + z^2 = 16## and ##x^2+y^2+(z-1^2) = 1##. The density of B is equal to the distance from its base, which is ##z = 0##. The mass of the solid B is ##\frac{188\pi}{3}##. Find the...
  24. Draconifors

    Triple integral using cylindrical coordinates

    Homework Statement The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
  25. karush

    MHB 15.5.63 - Rewrite triple integral in spherical coordinates

    Write interated integrals in spherical coordinates for the following region in the orders $dp \, d\theta \, d\phi$ and $d\theta \, dp \, d\phi$ Sketch the region of integration. Assume that $f$ is continuous on the region \begin{align*}\displaystyle...
  26. Moayd Shagaf

    I Difference Between d3x and triple Integral

    So My Question Is Simple, But It confuse me too much! What Is Difference between the notation ∫d3x and ∫∫∫dxdydz ?
  27. AutumnWater

    I Q about finding area with double/volume with triple integral

    So when finding the Area from a double integral; or Volume from a triple integral: If the curve/surface has a negative region: (for areas, under the x axis), (for volumes, below z = 0 where z is negative) What circumstances allow the negative regions to be taken into account as positive when...
  28. Anshul23

    Highschool graduate dealing with a triple integral?

    I recently came across a problem in Irodov which dealt with the gravitational field strength of a sphere. Took some time to get my head around it and figure how to frame a triple integral, but it felt good at the end. Am I going to start seeing triple integrals in the freshman year tho? If so...
  29. harpazo

    MHB How do I evaluate this triple integral for 2ze^(-x^2) over the given bounds?

    Evaluate the triple integral. Let S S S = triple integral The function given is 2ze^(-x^2) We are integrating over dydxdz. Bounds pertaining to dy: 0 to x Bounds pertaining to dx: 0 to 1 Bounds pertaining to dz: 1 to 4 S S S 2ze^(-x^2) dydxdz S S 2yze^(-x^2) from y = 0 to y = x dxdz S...
  30. harpazo

    MHB How to Set Up a Triple Integral for Volume Calculation?

    Use a triple integral to find the volume of the solid bounded by the graphs of the equations. x = 4 - y^2, z = 0, z = x I need help setting up the triple integral for the volume. I will do the rest.
  31. P

    MHB What is the value of the triple integral for the given solid and region?

    Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.
  32. toforfiltum

    Integrating triple integral over region W

    Homework Statement $$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##. Homework EquationsThe Attempt at a Solution Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have...
  33. T

    Mass of Region Bounded by y=sin(x), z=1-y, z=0, and x=0

    Homework Statement On a sample midterm for my Calc 3 class the following question appears: Find the mass of (and sketch) the region E with density ##\rho = ky## bounded by the 'cylinder' ##y =\sin x## and the planes ##z=1-y, z=0, x=0## for ##0\le x\le\pi/2##. Homework Equations $$ m= \int_{E}...
  34. R

    Where did I go wrong in my setup for this triple integral problem?

    Hey guys I've been working some triple integration problems and I've stumbled across a question that I'm having problems with So from the picture below my solution is incorrect and I can't seem to figure out where I went wrong. Is my setup for the integrals correct or is that where I've made my...
  35. nysnacc

    Triple integral forming a solid region

    Homework Statement Homework Equations Fubini's theorum The Attempt at a Solution I drawn the diagram with the limits (for x, y, and z) and come up with something with 4 faces, 5 corners, 8 edges is that something you guys got? Thanks
  36. T

    I Why this triple integral equals zero?

    Hello everyone, I have the this inquiry: if I compute de following integral: http://micurso.orgfree.com/Picture1.jpg by numerical methods I get cero as a result. I used Maxima and Mathematica and their functions for numerical integration give me an answer equal to cero. But, if I apply...
  37. N

    Triple integral in cylindrical coordinates

    1. Homework Statement I am trying to solve a triple integral using cylindrical coordinates. This is what I have to far . But I think I have choosen the limits wrong. Homework EquationsThe Attempt at a Solution [/B]
  38. defaultusername

    What is the mass of ceramic on the wire with non-uniform coating?

    Homework Statement A metal wire is given a ceramic coating to protect it against heat. The machine that applies the coating does not do so very uniformly. The wire is in the shape of the curve The density of the ceramic on the wire is Use a line integral to calculate the mass of the...
  39. C

    Triple integral in polar coordinate

    Homework Statement why x is p(cosθ)(sinφ) ? and y=p(sinθ)(cosφ)? z=p(cosφ) As we can see, φ is not the angle between p and z ... Homework EquationsThe Attempt at a Solution
  40. G

    MHB Evaluate this triple integral for calculating volume

    I'm tring to find this volume using a triple integral in the form $dy$ $dx$ $dz$ However I think I'm evaluating the wrong integral because the result is 1 when the volume should be 1/6... Can someone help me find out what I'm doing wrong? The set is $$V=\{(x,y,z)\in \mathbb{R^3}: x+y+2z...
  41. TheSodesa

    A sphere with a hole through it (a triple integral).

    Homework Statement A sphere has a diameter of ##D = 2\rho = 4cm##. A cylindrical hole with a diameter of ##d = 2R = 2 cm## is bored through the center of the sphere. Calculate the volume of the remaining solid. (Spherical or cylindrical coordinates?) hint: Place the shape into a convenient...
  42. T

    I How Is the Divergence Theorem Applied to Derive Vector Field Identities?

    In the image attached to this post, there is an equation on the top line and one on the bottom line. In the proof this image was taken from, they say this is a consequence of divergence theorem but I'm not quite understanding how it is. If anyone could explicitly explain the process to go from...
  43. TheSodesa

    Upper and lower bounds of a triple integral

    Homework Statement Let ##T \subset R^3## be a set delimited by the coordinate planes and the surfaces ##y = \sqrt{x}## and ##z = 1-y## in the first octant. Write the intgeral \iiint_T f(x,y,z)dV as iterated integrals in at least 3 different ways. Homework Equations \iiint_T f(x,y,z)dV =...
  44. W

    Volume of a Pond (Triple Integral)

    Homework Statement A circular pond with radius 1 metre and a maximum depth of 1 metre has the shape of a paraboloid, so that its depth z is z = x 2 + y 2 − 1. What is the total volume of the pond? How does this compare with the case where the pond has the same radius but has the shape of a...
  45. G

    Another triple integral problem

    Homework Statement Calculate \int_D \frac{dxdydz}{\sqrt{x^2+y^2+(z-A)^2}}, \: A>R on ## D = {(x,y,z)\: s.t. x^2+y^2+z^2 \leq R^2}##. Homework Equations In spherical coordinates: x=\rho cos\theta sin\phi\\y=\rho sin\theta sin\phi\\ z=cos\phi\\dxdydz=\rho^2sin\phi d\theta d\rho d\phi The...
  46. G

    How to solve a triple integral problem using cylindrical coordinates?

    Homework Statement Given E = [(x,y,z) s. t. 0 \leq x \leq 2, 0 \leq y \leq \sqrt{2x - x^2}, 0 \leq z \leq 2] Calculate \int_E z^3\sqrt{x^2+y^2}dxdydz Homework Equations In cylindrical coordinates: x=rcos(\theta)\\y=rsin(\theta)\\z=z\\dxdydz = \rho d\rho d\theta dz The Attempt at a...
  47. P

    Volume of ice cream cone triple integral

    Homework Statement Find the triple integral for the volume between a hemisphere centred at ##z=1## and cone with angle ##\alpha##.The Attempt at a Solution What I tried to do first was to get the radius of the hemisphere in terms of the angle ##\alpha##. In this case the radius is ##\tan...
  48. physkim

    Volume integral of a function over tetrahedron

    Homework Statement Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$ over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$ Homework Equations I was able to solve it mathematically, but still can't figure out why the answer is so small. I only understand...
  49. T

    Triple integral changing order of integration

    Homework Statement rewrite using the order dx dy dz \int_0^2 \int_{2x}^4\int_0^{sqrt(y^2-4x^2)}dz dy dx The Attempt at a Solution I am having trouble because i don't know what the full 3 dimensional region looks like but the part on the xy plane is a triangle bounded by x = 0 , y = 4 and y =...
  50. C

    Evaluate the triple integral paraboloid

    Evaluate the triple integral ∫∫∫E 5x dV, where E is bounded by the paraboloid x = 5y2+ 5z2 and the plane x = 5. My work so far: Since it's a paraboloid, where each cross section parallel to the plane x = 5 is a circle, cylindrical polars is what I used, so my bounds are 5y2+5z2 ≤x ≤ 5 ----->...
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