11 Triple Integrals

Question 1

What is the amount of each slice in a Cartesian triple integral?

Answer 1

sigma(x,y,z) dV which is volume density X volume

Question 2

How to integrate with respect to x,y, or z first in Cartesian triple integrals?

Answer 2

1. Identify the bounds for the first variable top and bottom boundary surfaces f1(other, variables) and f2(other, variables) 2. The second boundary will be projected onto the remaining v1 v2 plane, and will be g1(v3) to g2(v^3) then sigma(x,y,z) v1v2v3

Question 3

What do you put next to the integral lines in 1D, 2D and 3D integration?

Answer 3

1D: integral over interval I (linear density sigma(x)). 2D: integral over domain D (area density sigma(x,y)). 3D: integral over region W (volume density sigma(x,y,z)).

Question 4

How to write a triple integral in cylindrical form?

Answer 4

1. Write the bounds for region in terms of radial distance r, angle theta, and height above xy-plane z. 2. Write density function in terms of cylindrical coordinates (may need to convert). 3. Write infinitesimal volume dV = rdrdthetadz.

Question 5

What are the 4 equations to convert Cartesian to spherical?

Answer 5

X = psin(phi)cos(Theta). Y = psin(phi)cos(theta). Z = pcos(theta). P^2 = x^2 + y^2 + z^2

Question 6

What is the infinitesmal volume element in spherical?

Answer 6

DV = p^2 sin(phi)dpdthetadphi