Lecture 16
Applications of Integration
APMA E2000
Drew Youngren dcy2@columbia.edu
Announcements
- HW 9 due Tues.
-
Quiz 6 this week.
- Double/triple integrals
- Coordinate systems
-
Exam 2 Thurs, 04/03
- Same format as last time
- New seat assignments
- Not cumulative in the narrow sense but all math is cumulative.
1-minute review
Other Coordinates
- Polar coordinates $(r,\theta)$ \[ \iint_\mathcal R f(x,y)\,dxdy = \iint_\mathcal R f(r\cos \theta, r \sin \theta)\,r\,dr\,d\theta \]
Other Coordinates
- Cylindrical coordinates $(r,\theta, z)$\[ \iiint_\mathcal E f(x,y,z)\,dxdydz = \iiint_\mathcal E f(r\cos \theta, r \sin \theta,z)\,r\,dz\,dr\,d\theta \]
Other Coordinates
- Spherical coordinates $(\rho, \phi, \theta)$ \[ \iiint_\mathcal E f(x,y,z)\,dxdydz = \]\[\iiint_\mathcal E f(\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi ) \, \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \]
where the bounds of integration for $\mathcal R$ or $\mathcal E$ are translated appropriately.
Example
Find the volume of the "stadium" defined as the region above the $xy$-plane bound by a sphere of radius 2, a cylinder of radius 4, and the cone $x^2 + y^2 = z^2$, cut as shown.
\[ \int_{\pi/2}^{2\pi} \int_{\pi/4}^{\pi/2}\int_2^{4\csc \phi} \rho^2 \sin\phi \,d\rho\,d\phi\,d\theta = \]
\[ = (32 - 2\sqrt{2})\pi \]
\[ \spadesuit = \int d\spadesuit \]
Applied Integration
The whole is the sum of its parts.
Density
Density, a rate of $\frac{\text{mass}}{\text{volume}}$ can vary continuously as a function of location $\mu(x,y,z)$. \[ \text{mass}_{\text{total}} = \iiint\limits_{\mathcal E} dm = \iiint\limits_{\mathcal E} \mu \,dV \]
but density can be more general: $\frac{\text{stuff}}{\text{space}}$.
Examples
- Resistivity $\mu(x)$ along a wire, $\frac{\Omega}{{\rm m}}$. \[\Omega = \int_0^\ell \mu(x)\, dx\]
- Probability density function $\mu(x,y)$ of two random variables. \[P(E) = \iint\limits_E \mu(x,y)\, dA\]
- Concentration of chemicals, like $[\text{H}_2\text{CO}_3] = \mu(x,y,z)$. \[\text{total carbonic acid} = \iiint\limits_D \mu(x,y,z)\,dV\]
Center of Mass
Discrete Mass Distribution
Suppose point masses $m_1$ and $m_2$ are located at positions $x_1$ and $x_2$, resp., along a line. The center of mass $\bar x$ is the position that balances the "moments".
\[m_1 (x_1 - \bar x) + m_2(x_2 - \bar x) = 0\]
Discrete Mass Distribution
Suppose point masses $m_1$ and $m_2$ are located at positions $x_1$ and $x_2$, resp., along a line. The center of mass $\bar x$ is the position that balances the "moments".
\[\bar x = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}\]
Discrete Mass Distribution
Suppose point masses $m_1$ and $m_2$ are located at positions $x_1$ and $x_2$, resp., along a line. The center of mass $\bar x$ is the position that balances the "moments".
\[\bar x = \frac{m_1}{m_1 + m_2} x_1 + \frac{m_2}{m_1 + m_2} x_2\]
aka, a weighted average of the positions.
Continuous Mass Distribution
Suppose mass is distributed continuously along the interval $a \leq x \leq b$ with density $\mu(x)$. The center of mass $\bar x$ is the position that balances the "moments".
\[\int_a^b (x - \bar x) \mu(x)\,dx = 0\]
Continuous Mass Distribution
Suppose mass is distributed continuously along the interval $a \leq x \leq b$ with density $\mu(x)$. The center of mass $\bar x$ is the position that balances the "moments".
\[\bar x = \frac{\int_a^b x \mu(x)\,dx}{\int_a^b \mu(x)\,dx} \]
aka, a weighted average of the positions.
Higher Dimensions
A solid region $\mathcal E$ with continuously varying density (mass per unit volume) $\mu(x,y,z)$ has total mass \[M = \iiint_\mathcal E \mu(x,y,z)\,dV.\]
The center of mass $(\bar{x},\bar y,\bar z)$ is the (weighted) average position of the mass in the object. More concisely, \[\bv{\bar{x} \\ \bar{y} \\ \bar{z}} = \frac1M \bv{\iiint_\mathcal E x \mu(x,y, z)\,dV \\ \iiint_\mathcal E y \mu(x,y, z)\,dV \\ \iiint_\mathcal E z \mu(x,y, z)\,dV}\]
Example
The nose of a race car is modeled by half of a right cone with height $h$ and radius $R$. Assuming uniform density, how high is the center of mass from the bottom.
Moment of Inertia
Concept
CoM measures where mass is located on average. It can't measure how far it is spread out.
The moment of inertia does this. For a mass distribution $\mu(x,y,z)$ and a particular axis of rotation, \[ I = \iiint\limits_{\mathcal E} (\text{distance to axis})^2 \mu(x,y,z)\,dV \]
Example
Find the center of mass and the moments of inertia about the standard axes for a mass distribution $\mu$ on the unit square in $xy$.
\[\mu(x,y) = x + 2 y^2 \] in units of mass/area.
\[\bar{x} = \frac{\int_0^1\int_0^1 x (x + 2y^2)\,dy\,dx}{\int_0^1\int_0^1 (x + 2y^2)\,dy\,dx} = \frac47\]
\[\bar{y} = \frac{\int_0^1\int_0^1 y (x + 2y^2)\,dy\,dx}{\int_0^1\int_0^1 (x + 2y^2)\,dy\,dx} = \frac{9}{14}\]
\[I_{x} = \int_0^1\int_0^1 y^2 (x + 2y^2)\,dy\,dx = \frac{17}{30}\]
\[I_{y} = \int_0^1\int_0^1 x^2 (x + 2y^2)\,dy\,dx = \frac{17}{36}\]
MoI Examples
Find an expression for the moment of inertia $I$ (about a central axis) in terms of the total mass $M$ (with uniform density) for the following shapes:
- Cube with side $2R$.
- Right cone with base radius $R$ and height $h$.
- Half-ball with radius $R$.
Learning Outcomes
You should be able to...
- Interpret a density function (in various dimensions) as a distribution or an indication of "where stuff is."
- Correctly compute center of mass for various distributions, using symmetric arguments where appropriate.
- Reason about and compare moments for different distributions without necessarily computing the numeric value.
- Reinterpret basic probability questions about multiple random variables in terms of integrals.