Calculus II Lesson 5: Partial Fractions, Center of Mass

1. Quiz
2. Homework questions?
3. Upcoming
   1. Problem Presentation
   2. Questions?
4. Partial Fractions
   1. Partial Fractions
   2. Exercises
   3. More general
5. Center of Mass
   1. Moments, 1D
   2. Moments, 2D
   3. Moments, 2D
   4. Center of Mass
   5. Centroid of Rectangle
   6. Centroid of Curve
   7. Moments of one rectangle
   8. Example

Quiz

Homework questions?

• HW 1 due today.
• Exit ticket due today also.

Upcoming

• Today: partial fractions, center of mass application
• Thursday: review for exam 1
• Exam 1: Monday, 2/16.
  • Chapter 1
  • Sections 3.1, 3.2, 3.4
  • Section 2.6
• Problem Presentation 1: Monday, Feb 23, Thursday, Feb 26, Monday, March 2.

Problem Presentation

• Describe a challenging problem
  • Homework (textbook / MOM)?
  • Textbook, not assigned?
  • Anywhere else?
• Explain the solution
  • Pretend you’re working through a problem “on the board” in class.
  • What formulas / techniques / etc were needed?
  • How did you know which method to use?
• Use either slides or can go over the problem on the chalkboard.
• Keep it short: roughly 5 minutes.
• See rubric on BrightSpace

Questions?

• Meaning of Definite Integral / FTC
• Integration by substitution
• Exponential / Logarithmic Functions
• Inverse Trig
• Integration by Parts
• Trig powers
• Products of trig functions of different angles

Partial Fractions

How do we compute the following?

\[\int \frac{1}{x^2 - 1} dx\]

First, notice that: \(\frac{1/2}{x-1} - \frac{1/2}{x+1} = \frac{1}{x^2 - 1}\)

(Algebra! Get commone denominators on the left side)

Then: $\int (\frac{1/2}{x-1} - \frac{1/2}{x+1})dx = \frac{1}{2}(\ln|x-1| - \ln|x+1|) + C$

So the question becomes: how do we go from $\frac{1}{x^2 - 1}$ to $\frac{1/2}{x-1} - \frac{1/2}{x+1}$?

Another example:

\[\frac{1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}\]

How would I solve for $A$ and $B$?

\[\begin{align} \frac{1}{(x+1)(x+2)} &= \frac{A}{x+1} + \frac{B}{x+2} \\ &= \frac{A(x+2) + B(x+1)}{(x+1)(x+2)} \end{align}\]

• $A(x+2) + B(x+1) = 1$, no matter what $x$ is
• $x = -2$, then $B = -1$.
• $x = -1$, then $A = 1$.

So: $\frac{1}{(x+1)(x+2)} = \frac{1}{x+1} - \frac{1}{x+2}$

Partial Fractions

This is the method of partial fractions.

• Rational function $\frac{P(x)}{Q(x)}$
• Factor $Q(x)$
• For most examples we will do: $Q(x) = (ax + b)(cx + d)$
• Then $\frac{P(x)}{Q(x)} = \frac{A}{ax + b} + \frac{B}{cx + d}$ is the partial fraction decomposition
• How to find $A$ and $B$?
  • Common denominator
  • Pick special values of $x$ that cancel nicely.
  • Use zeroes of $Q(x)$

Exercises

Decompose into partial fractions and integrate.

1. $\int \frac{1}{(x+2)(x-1)} dx$
2. $\int \frac{1}{x^2 - 6x + 5} dx$

Solutions:

\[\int \frac{1}{(x+2)(x-1)} dx\]

• Using partial fractions, $\frac{1}{(x+2)(x-1)} = \frac{1/3}{x+2} - \frac{1/3}{x-1}$.
• $\int \frac{1/3}{x+2} dx - \int \frac{1/3}{x-1} dx$
• $\frac{1}{3} \ln|x+2| - \frac{1}{3} \ln|x-1| + C$

\[\int \frac{1}{x^2 - 6x + 5} dx\]

• Factor: $x^2 - 6x + 5 = (x-5)(x-1)$
• Partial Fractions: $\frac{1}{(x-5)(x-1)} = \frac{1/4}{x-5} - \frac{1/4}{x-1}$
• $\int \frac{1/4}{x-5} dx - \int \frac{1/4}{x-1} dx$
• $\frac{1}{4} \ln|x-5| - \frac{1}{4} \ln|x-1| + C$

More general

\[\frac{1}{(x+1)(x+2)(x+3)} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{x+3}\]

• $x = -1$: $A(x+2)(x+3) = 1$ means $A = \frac{1}{2}$.
• $x = -2$: $B(x+1)(x+3) = 1$ means $B = -1$.
• $x = -3$: $C(x+1)(x+2) = 1$ means $C = \frac{1}{2}$.

Then we can just integrate:

$\frac{1}{2}\int \frac{dx}{x+1} - \int \frac{dx}{x+2} + \frac{1}{2} \int \frac{dx}{x+3}$

Final answer: $\frac{1}{2}\ln|x+1| - \ln|x+2| + \frac{1}{2}\ln|x+3| + C$

Center of Mass

Suppose:

• $m_1 = 10$ kg
• $m_2 = 20$ kg
• $x_1 = -2$
• $x_2 = 2$

What is the balancing point?

Moments, 1D

• Looking for $x$ such that $mass \cdot distance$ is the same on both sides.
• $20(2-x) = 10(x - (-2))$
• $40 - 20x = 10x + 20$
• $20 = 30x$
• $x = \frac{20}{30} = \frac{2}{3}$

30 is the total mass of the system. 20 is the moment of the system with respect to the origin.

Moments, 2D

• $M_x$ = moment with respect to $x$-axis? $m_1 \cdot y_1$ (mass times distance to x-axis)
• $M_y$ = moment with respect to $y$-axis? $m_1 \cdot x_1$ (mass times distance to y-axis)

Moments, 2D

Center of mass? Sum of all the moments:

• $M_x = 10(1) + 5(-3) = -5$
• $M_y = 10(1) + 5(4) = 30$
• Total mass: 15kg.
• Center of mass?
  • $(\frac{M_y}{M}, \frac{M_x}{M})$
  • $(2, -\frac{1}{3})$

Center of Mass

• Pretend all mass is concentrated on this one point
• Useful for calculating gravity
  • Much harder to compute the forces on all the individual particles

Centroid of Rectangle

• Rectangle of uniform density $\rho$?
• $(\frac{x_1 + x_2}{2}, \frac{y}{2})$

Centroid of Curve

• Assume uniform density $\rho$.
• Approximate the moments and total mass by rectangles.
• Split into $n$ rectangles
• Endpoints $x_0, x_1, \ldots, x_n$

Moments of one rectangle

• $M_x$ = (mass of rectangle) $\cdot \frac{y}{2}$
• $M_y$ = (mass of rectangle) $\cdot x_i$ (if $x_i$ and $x_{i+1}$ are close)
• mass of rectangle $= \rho y \Delta x$

Let’s find the moments:

• $M_x \approx \sum\limits_{i = 1}^n \rho (y \Delta x) (\frac{y}{2})$
• $M_y \approx \sum\limits_{i = 1}^n \rho (y \Delta x) (x)$
• $m \approx \sum\limits_{i = 1}^n \rho y \Delta x$

Let’s take the limit. As $n \rightarrow \infty$, if $y = f(x)$:

• $M_x = \rho \int_a^b \frac{(f(x))^2}{2} dx$
• $M_y = \rho \int_a^b xf(x) dx$
• $m = \rho \int_a^b f(x) dx$
• Center of mass = $(\frac{M_y}{m}, \frac{M_x}{m})$
• $\rho$ cancels!

Example

• Region enclosed by $y = \sin(x)$
• $x = 0$ to $x = \pi$
• Center of mass: $(\pi/2, \pi/8)$
• Confirm this!