Chapter 7
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1
What is the general form of a triple integral of f(x, y, z)?
$\iiint_V f(x, y, z) \, dV$
2
What are the steps to fill in?
1. Sketch V
2. Decide on dV
3. Find inner limits (draw ab and ac plane)
4. Find middle limits (draw cb plane)
5. Find outer limits (draw ca or cb plane)
2. Decide on dV
3. Find inner limits (draw ab and ac plane)
4. Find middle limits (draw cb plane)
5. Find outer limits (draw ca or cb plane)
3
What are the three options for dV in triple integrals?
dx dy dz - flat 'slices'
dy dx dz
dx dz dy horizontal 'slices' (slices cut y axis)
dz dx dy
dy dz dx vertical 'slices (slices cut x axis)
dz dy dx
dy dx dz
dx dz dy horizontal 'slices' (slices cut y axis)
dz dx dy
dy dz dx vertical 'slices (slices cut x axis)
dz dy dx
4
What is the formula for the volume of V in triple integrals?
$\iiint_V dV = \text{volume of } V$
5
$\text{What are the formulas for } \bar{x}, \bar{y}, \bar{z} \text{ in double integrals}?$
$\bar{x} = \frac{\iiint_V x \, dV}{\iiint_V \, dV}, \quad
\bar{y} = \frac{\iiint_V y \, dV}{\iiint_V \, dV}, \quad
\bar{z} = \frac{\iiint_V z \, dV}{\iiint_V \, dV}$
6
$\text{What are the conversions for triple integrals in cylindrical? } (x, y, z, dV)$
$dV = R \, dR \, d\phi \, dz$
7
$\text{What does the order of integration } dR \, d\phi \, dz \text{ represent in cylindrical triple integrals?}$
8
$\text{What does the order of integration } dz \, d\phi \, dR \text{ represent in cylindrical triple integrals?}$
9
$\text{What are the conversions for triple integrals in spherical? } (x, y, z, dV)$
10
$\text{What does the order of integration } dR \, d\theta \, d\phi \text{ represent in spherical triple integrals?}$
11
What is the Jacobian J for a coordinate change from (x, y, z) --> (u, v, w) in triple integrals?
$\mathbf{J} = \det
\begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\
\frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}
\end{bmatrix}$
$dx \, dy \, dz = |\mathbf{J}| \, du \, dv \, dw$
$du \, dv \, dw = \frac{dx \, dy \, dz}{|\mathbf{J}|}$
$dx \, dy \, dz = |\mathbf{J}| \, du \, dv \, dw$
$du \, dv \, dw = \frac{dx \, dy \, dz}{|\mathbf{J}|}$
12
What are the functions for the coordinate change for an elipsoid?
$x = r \sin\theta \cos\phi, \quad y = r \sin\theta \sin\phi, \quad z = r \cos\theta$