Calculus 3 Triple Integration in Spherical Coords

In summary, the conversation discusses using spherical coordinates to evaluate a triple integral for a solid between two spheres. The boundaries and final answer are mentioned, with the expert providing an explanation for the answer being zero. The expert also suggests using a different range for the variable rho.
  • #1
Wargy
3
0

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 I think. Currently I am using 0 to 2[tex]\pi[/tex] for [tex]\vartheta[/tex], 0 to [tex]\pi[/tex] for [tex]\varphi[/tex] and 1 to 2 for [tex]\rho[/tex]. When solving, I get zero as my final answer, and since I'm not clear on the conceptual meaning of a triple integral that isn't of a function that equals 1 (volume) I don't know if this answer makes sense.
 
Last edited:
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  • #2
It is zero. It's the integral of z on the volume between two spheres centered on the origin. It's as much positive as negative. The two cancel. But you should be using rho from 1 to 2.
 
  • #3
Nevermind, read what you wrote again.
Thanks for the quick reply!
 
Last edited:
  • #4
Makes a lot of sense.
 

Related to Calculus 3 Triple Integration in Spherical Coords

1. What is Calculus 3 Triple Integration in Spherical Coords?

Calculus 3 Triple Integration in Spherical Coords is a mathematical technique used in vector calculus to solve problems involving three-dimensional shapes and volumes. It involves using spherical coordinates, which represent points in three-dimensional space using a radial distance, an angle in the xy-plane, and an angle above the xy-plane.

2. What is the difference between spherical coordinates and Cartesian coordinates?

Spherical coordinates use a different system of coordinates compared to Cartesian coordinates, which use x, y, and z coordinates to represent points in three-dimensional space. Spherical coordinates use a radial distance, an angle in the xy-plane, and an angle above the xy-plane to represent points in three-dimensional space.

3. How do you convert between spherical and Cartesian coordinates?

To convert from spherical coordinates to Cartesian coordinates, you can use the following equations:
x = ρsinφcosθ
y = ρsinφsinθ
z = ρcosφ
where ρ represents the radial distance, φ represents the angle in the xy-plane, and θ represents the angle above the xy-plane. To convert from Cartesian coordinates to spherical coordinates, you can use the following equations:
ρ = √(x^2 + y^2 + z^2)
φ = tan^-1(y/x)
θ = cos^-1(z/ρ)

4. What types of problems can be solved using Calculus 3 Triple Integration in Spherical Coords?

Calculus 3 Triple Integration in Spherical Coords can be used to solve problems involving three-dimensional shapes and volumes, such as finding the volume of a sphere or calculating the mass of a solid with varying density. It can also be used to solve problems in physics and engineering, such as calculating electric fields or gravitational forces.

5. What are some common applications of Calculus 3 Triple Integration in Spherical Coords?

Calculus 3 Triple Integration in Spherical Coords is commonly used in fields such as physics, engineering, and astronomy. It can be used to solve problems involving spherical objects, such as planets and stars, and to analyze physical phenomena in three-dimensional space. It is also used in computer graphics and 3D modeling to represent and manipulate three-dimensional objects.

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