# What is Triple integration: Definition and 36 Discussions

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. ### How to find the limits of a volume integral?

Homework Statement If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##. The Attempt at a Solution From gauss divergence theorem we know ##\int...
2. ### Triple integration - find volume

Homework Statement Homework Equations in the pic The Attempt at a Solution why is there an extra "r" in the highlighted line? my attempt: ## \int_0^{2\pi}\int_0^2\int_{r^2}^41\ dy\ r\ dr\ d\theta ## = ##\int_0^{2\pi}\int_0^2\left(4-r^2\right)\ r\ dr\ d\theta## thanks
3. ### Triple Integration: Solve Homework Equation

Homework Statement I'm trying to figure out the other parameters to solve the problem Homework Equations I know sqrt(x^2+y^2+z^2) = p The Attempt at a Solution I changed the integrand to p^3 sin(theta) since p * p^2 sin(theta) Then for the first integration sign, I know how to get the...
4. ### Triple integration question

Homework Statement Shown in the photo attached. 2. Homework Equations ∫V r2Sinθdθdφdr in spherical coordinates ∫V dxdydz in cartesian coordinates equation of a sphere x2+y2+z2=r2 The Attempt at a Solution In this case y=(y-2): sphere displaced on the y-axis. and since it is bound by all...
5. ### Triple integration over portion of Sphere

Homework Statement Homework Equations spherical Jacobean The Attempt at a Solution I have (sorry, have to capture my work, too hard to type) then the integration of p3 ep2 = 1/2 ep2 (p2-3/2) ??
6. ### Changing Order of Triple Integration

Homework Statement \int_0^5 \int_0^2 \int_0^{4-y^2}\ \, dxdydx Change order to dydxdz Homework EquationsThe Attempt at a Solution I'm confused mainly because the limits are mostly numbers, not functions. I graphed the limits in @D and #d and this is what I got: \int_0^{4-y^2} \int_0^5...
7. ### Defining rho in spherical coordinates for strange shapes?

Homework Statement The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere...
8. ### How to determine the limits for triple integration?

Homework Statement Evaluate the triple integral: ∫ x dxdydz A where A = {(x; y; z) : x, y, z > 0, x + y + z ≤ 1} . Homework Equations None that I know of. The Attempt at a Solution The problem I have is determining the limits for x, y and z. I don't really understand the following...
9. ### Cone with spherical top triple integration

Homework Statement Homework Equations ∫∫∫dV The Attempt at a Solution Ok so I started by setting my bounds equal to √(200-x^2-y^2) ≥ z ≥ √(x^2+y^2), √(100-x^2) ≥ y ≥ -√(100-x^2), 10 ≥ x ≥ -10 which I got from solving z^2 = (200-x^2-y^2) = x^2+y^2 => x^2+y^2 = 100 but it...
10. ### Multivariable Calculus Triple Integration Problem

Homework Statement Express the iterated integral ∫[0,1]∫[0,1-y^2]∫[0,y] f(x,y,z)dzdxdy a. as a triple integral (i.e., describe the region of integration); b. as an iterated integral in the order z, y, x; c. as an iterated integral in the order y, z, x: The Attempt at a Solution so...
11. ### Triple Integration from Rectangular to Spherical Coordinates

Homework Statement Convert the integral from rectangular coordinates to spherical coordinates 2 √(4-x^2) 4 ∫ ∫ ∫ x dz dy dx -2 -√(4-x^2) x^2+y^2 Homework Equations x=ρ sin∅ cosθ y=ρ sin∅ cosθ z=ρ cos∅ In case the above integrals cannot be understood: -2...
12. ### Triple Integration of charge density question

Homework Statement a)Calculate the total charge a square capacitor plate would have with width x, height y, thickness z, and charge density f(x,y,z) = 1+x+y b)Calculate the total charge a sphere would have with radius r, and charge density f(x,y,z)=x+y+z Use the triple integration seen in...
13. ### Triple Integration for Volume

Homework Statement Find the volume of the solid in the first octant bounded by the graphs of: z=1-y2 y=2x x=3 Homework Equations I was able to graph all three but I can't picture the region for integration. I'm not sure if I even have to graph it or if I can get my limits without the graph...
14. ### Volume of Frustum Using Triple Integration

Volume of Frustum Using Triple Integral [Solved] Homework Statement Edit: I've solved the issue! My limits of r were wrong. Instead of this: V=\int_{z=0}^{\frac{h}{2}}\int_{r=0}^{\frac{R}{h}z+R}\int_{\theta=0}^{2\pi}r\ d\theta\ dr\ dzIt should have been this...
15. ### Triple integration in spherical polars

Homework Statement Determine the value of \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2-y^2}} \sqrt{x^2+ y^2 + z^2} dz dy dx The Attempt at a Solution So in spherical polars, the integrand is simply ρ. \sqrt{1- x^2- y^2} = z = ρ\cos\phi = \cos\phi since we are on the unit...
16. ### Triple integration to find volume of regions

Homework Statement http://s2.ipicture.ru/uploads/20111231/kczcXUuF.jpg The attempt at a solution So, I'm using the transformation to spherical coordinates (ρ,∅,θ) Description of region: For θ and ∅ fixed, ρ varies from 0 to 4. For θ fixed, ∅ varies from 0 to ∏. (i suspect the error...
17. ### Change the order of triple integration

Homework Statement Find all 5 other orders of intergration Homework Equations \int_{0}^{1}\int_{0}^{x^2}\int_{0}^{y}dzdydx The Attempt at a Solution I am really confused as to how to represent it graphicaly so I don't have any visuals to help. Can you guys help me? Thank you SO...
18. ### Triple Integration: Evaluating by Changing Order of Integral

Homework Statement Evaluate the integral by changing the order of the integration in an appropriate way. ∫∫∫ ze-(y2+z2)dzdydx z goes from 0 to ∞, y goes from x/6 to 3, x goes from 0 to 18 Homework Equations The Attempt at a Solution to change the integration ∫∫∫...
19. ### Triple integration problem

Find the volume lying below z = 3 - 2y and above z = x^2 + y^2. How would I go about finding the limits of integration for this problem?
20. ### Need help with triple integration problem

Homework Statement \int _0^{\sqrt{\pi }}\int _0^x\int _0^{x z}x^2 \text{Sin}[y]dydzdx
21. ### Changing the order of a triple integration

I'm given this definite integral: \int_0^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)dzdydx I need to change the order to dydxdz, but I'm stuck trying to get the limits of integration wrt y. \int_0^{1}\int_{x^2}^{0}\int_{}^{}f(x,y,z)dydzdx How do I find the limits of y?
22. ### Change the order of triple integration

Homework Statement Rewrite \int_{0}^{2}\int_{0}^{y^3}\int_{0}^{y^2}dzdxdy as an integral with order dydzdx. Homework Equations N/A The Attempt at a Solution Honestly, I got as far as sketching it: and after sketching it, I'm lost... I can't figure out how to set z or y, but...
23. ### Volume of 2-sphere using triple integration (rect. cord)

Homework Statement Can anyone help me with the volume of a 2-sphere in rect cordinates? I'm having problems with the limits of the triple integral. Ultimately I will need to go beyond the 2-sphere to a 3 and 4 using quadruple and five integrals respectively. Radius at r from 0 vector. Homework...
24. ### Finding a volume by triple integration

I need to find the volume between the cone z=sqrt(x^2+y^2) and the sphere x^2+y^2+z^2=1 that lies in the first octant. Now I've used cylindrical coordinates for this and found the limits to be 0<theta<pi/2 0<r<1/sqrt(2) r<z<sqrt(1-r^2) I've done the triple integral and found the answer...
25. ### Calculus 3 Triple Integration in Spherical Coords

Homework Statement Use spherical coordinates to evaluate the triple integral z dV where Q is the solid that lies between x^2+y^2+z^2=1 and x^2+y^2+z^2=4. Homework Equations Not sure what goes here :P The Attempt at a Solution I've gotten everything set up, I am having problems with boundaries...
26. ### Triple Integration Help

Homework Statement Sketch the solid B that lies inside the surface x^2 + y^2 = 1 and is bounded above and below by the surface x^2 + y^2 + z^2= 2^2. Then find the volume of B. Homework Equations projxy = projection onto the xy plane, proj zy = projection on the zy plane The Attempt...
27. ### Triple Integration of a Sphere in Cylindrical Coordinates

Homework Statement The problem was to find the volume enclosed by a sphere of radius "a" centered on the origin by crafting a triple integral and solving for it using cylindrical coordinates. Homework Equations x^{2}+y^{2}+z^{2}=a^{2} : Equation for a sphere of radius "a" centered on...
28. ### Triple Integration of a Strange Cylinder

Hey all, If anyone has some hints on how to do this one it would be much appreciated: Find the volume of the region given by x2 + y2 ≤ a2, 0 ≤ z ≤ x. So I've gone to cylindrical polars, and threw in the Jacobian, r. If I integrate with my bounds being: 0 to a, 0 to pi, 0 to...
29. ### Triple Integration: Transform Equation to Spherical Coordinates

Homework Statement Transform the equation from cartesians coordinates to spherical coordinates. Homework Equations \int_\infty\int_\infty\int_\infty exp\left\{ \frac{-\left| \vec{x'}-\vec{x}_0 \right|^2}{2 \sigma} \right\} \frac{\left( \vec{x} - \vec{x'} \right)}{\left| \vec{x} -...
30. ### Volume of a solid by triple integration

Homework Statement Find the volume of the solid inside the sphere x^2 + y^2 + z^2 = 4 and over the paraboloid 3z = x^2 + y^2 The Attempt at a Solution This should be easy to calculate using polar coordinates. The limits for z is [r^2/2, sqrt(4-r^2)] and for tetha: [0, 2*pi], but how do...
31. ### Triple integration w/spherical coordinates

[b]1. "Find the mass of part of the solid sphere x^2 + y^2 + z^ 2 ≤ 25 in the 1st octant x ≥ 0, y ≥ 0, z ≥ 0 where mass density is f (x, y, z ) = (x^2 + y^2 + z^2 )^3/2 ." [b]3. These problems are really stumping me! I need somebody to work it out/explain it to me! What will the limits...
32. ### Triple Integration: Evaluate & Check Answer

Homework Statement Evaluate the following integral by changing to cylindrical coordinates: I displayed the question and my attempt in the document attached. Homework Equations The Attempt at a Solution The attempt is in the document attached. Please help me to check whether I...
33. ### Volume of a tetrahedron using triple integration

Homework Statement Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1,0,0), (0,2,0) and (0,0,3) Homework Equations V=∫∫∫dV ...D The Attempt at a Solution I set up the problem as so: 1 -2x+2...-3x+3 ∫...
34. ### Help with triple integration problem

hey! i need some help with a triple integration problem using spherical coordinates. it's the volume of a small part of a sphere. rho from 5 to 6, phi from pi/6 to pi/4 and theta from pi/4 to pi/3. i got an answer of (-91/72) x pi x (sqrt(2)-sqrt(3))...am i right? Thanks!
35. ### Double, triple integration

alo, i ve done single variable integration at school but i m trying to understand some vector analysis and going through books like Schey's Div, Grad Curl and all that as well as Schaum s vector analysis and to be able to understand i need to know exactly what a double integral is...i ve...
36. ### Triple integration for volume

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 Can someone help find the triple integral to find the volume.