Integration of Rational Functions by Partial Fractions and Volumes Using Cross-Sections

Integration of Rational Functions by Partial Fractions

Introduction to Partial Fractions

Integration by partial fractions is a technique used to integrate rational functions, which are quotients of polynomials. This method decomposes a complex rational function into a sum of simpler fractions that are easier to integrate.

• Rational Function: A function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials.

• Proper Rational Function: The degree of $P(x)$ is less than the degree of $Q(x)$.

• Improper Rational Function: The degree of $P(x)$ is greater than or equal to the degree of $Q(x)$. In this case, perform polynomial long division first.

Method of Partial Fractions (Proper Case)

The decomposition depends on the factorization of the denominator $Q(x)$:

1. For each distinct linear factor $(x - r)$ of $Q(x)$, assign a sum of the form: $\frac{A_1}{x - r} + \frac{A_2}{(x - r)^2} + \cdots + \frac{A_m}{(x - r)^m}$

2. For each irreducible quadratic factor $x^2 + px + q$ (where $p^2 - 4q < 0$), assign: $\frac{B_1 x + C_1}{x^2 + px + q} + \frac{B_2 x + C_2}{(x^2 + px + q)^2} + \cdots$

3. Set the original function equal to the sum of all partial fractions and arrange terms by decreasing powers of $x$.

4. Equate coefficients of corresponding powers of $x$ and solve for the unknowns.

Example: Distinct Linear Factors

Given $\int \frac{2x^2 + 3x - 2}{x^3 + 3x^2 - 2x} dx$, factor the denominator and decompose:

• Factor: $x^3 + 3x^2 - 2x = x(2x - 1)(x + 2)$

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

• Solve for $A, B, C$ by equating coefficients.

Example: Irreducible Quadratic Factors

Given $\int \frac{x}{(x - 2)(x^2 + 1)(x^2 + 4)} dx$, the decomposition includes terms like $\frac{A}{x - 2}$, $\frac{Bx + C}{x^2 + 1}$, $\frac{Dx + E}{x^2 + 4}$.

Summary Table: Partial Fraction Decomposition Forms

Denominator Factor	Partial Fraction Form
Distinct Linear $(x - r)$	$\frac{A}{x - r}$
Repeated Linear $(x - r)^m$	$\frac{A_1}{x - r} + \cdots + \frac{A_m}{(x - r)^m}$
Irreducible Quadratic $x^2 + px + q$	$\frac{Bx + C}{x^2 + px + q}$
Repeated Irreducible Quadratic $(x^2 + px + q)^n$	$\frac{B_1 x + C_1}{x^2 + px + q} + \cdots + \frac{B_n x + C_n}{(x^2 + px + q)^n}$

Volumes Using Cross-Sections

Introduction to Solids of Known Cross-Section

The volume of a solid can be found by integrating the area of cross-sections perpendicular to an axis. This method is foundational in applications of definite integrals.

Volume of a Cylindrical Solid

For a solid with constant cross-sectional area $A$ and height $h$, the volume is $V = Ah$.

General Volume Formula

If the cross-sectional area varies with $x$, the volume from $x = a$ to $x = b$ is:

• $V = \int_a^b A(x) dx$

Steps for Calculating the Volume of a Solid

1. Sketch the solid and a typical cross-section.

2. Find a formula for $A(x)$, the area of a typical cross-section.

3. Find the limits of integration.

4. Integrate $A(x)$ to find the volume.

Cavalieri's Principle

If two solids have equal heights and equal cross-sectional areas at every level, then they have equal volumes.

Solids of Revolution: The Disk Method

Volume by Disks for Rotation About the x-Axis

When a region is revolved about the x-axis, the resulting solid's volume can be found using the disk method:

• $V = \int_a^b \pi [R(x)]^2 dx$

• Where $R(x)$ is the distance from the axis of rotation to the outer edge of the region.

Example: Volume of a Sphere

The volume of a sphere of radius $r$ can be derived by integrating the area of circular cross-sections:

• Cross-sectional area at $x$: $A(x) = \pi (r^2 - x^2)$

• Volume: $V = \int_{-r}^r \pi (r^2 - x^2) dx = \frac{4}{3} \pi r^3$

Solids of Revolution: The Washer Method

Volume by Washers for Rotation About the x-Axis

If the solid has a hole (i.e., the region does not border the axis of revolution), use the washer method:

• Outer radius: $R(x)$

• Inner radius: $r(x)$

• Area of washer: $A(x) = \pi [R(x)]^2 - \pi [r(x)]^2$

• Volume: $V = \int_a^b \pi ([R(x)]^2 - [r(x)]^2) dx$

Example: Region Between Two Curves

To find the volume generated by rotating the region between $y = x$ and $y = x^2$ about the x-axis:

• Outer radius: $R(x) = x$

• Inner radius: $r(x) = x^2$

• Volume: $V = \int_0^1 \pi (x^2 - x^4) dx$

Example: Rotation About a Line Other Than the Axis

When rotating about $y = 2$, adjust the radii accordingly:

• Outer radius: $R(x) = 2 - x^2$

• Inner radius: $r(x) = 2 - x$

• Volume: $V = \int_0^1 \pi ([2 - x^2]^2 - [2 - x]^2) dx$

Summary Table: Methods for Finding Volumes of Solids of Revolution

Method	Formula	When to Use
Disk	$V = \int_a^b \pi [R(x)]^2 dx$	Solid has no hole; region borders axis of revolution
Washer	$V = \int_a^b \pi ([R(x)]^2 - [r(x)]^2) dx$	Solid has a hole; region does not border axis
Cross-Section	$V = \int_a^b A(x) dx$	General solids with known cross-sectional area

Additional info: The notes above include both the algebraic and geometric foundations for integrating rational functions by partial fractions and for calculating volumes of solids using cross-sections, disks, and washers. These are core topics in Calculus II and are essential for applications in physics, engineering, and advanced mathematics.