Calculus II Lecture Notes

These lecture notes provide a comprehensive overview of the fundamental concepts covered in a standard Calculus II course. This course serves as a bridge between the basic differentiation and integration techniques of Calculus I and the multivariable calculus of Calculus III. The primary focus is on advanced integration techniques, infinite sequences and series, and the geometry of space.

1. Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It is used to integrate products of functions where substitution is not applicable. The formula is derived from the derivative of the product of two functions, $u$ and $v$.

Choosing $u$ and $dv$ correctly is crucial. A common mnemonic is LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential), which prioritizes the choice for $u$ from left to right.

2. Trigonometric Integrals

This section deals with integrals involving powers of sine, cosine, tangent, and secant. The strategy depends on the powers of the trigonometric functions.

Case 1: Powers of Sine and Cosine

• If the power of sine is odd ($m$ is odd), save one sine factor and convert the rest to cosines using $\sin^2 x = 1 - \cos^2 x$. Substitute $u = \cos x$.
• If the power of cosine is odd ($n$ is odd), save one cosine factor and convert the rest to sines using $\cos^2 x = 1 - \sin^2 x$. Substitute $u = \sin x$.
• If both powers are even, use the half-angle identities:
  $\sin^2 x = \frac{1}{2}(1 - \cos(2x))$
  $\cos^2 x = \frac{1}{2}(1 + \cos(2x))$

Case 2: Powers of Tangent and Secant

• If the power of secant is even, save a factor of $\sec^2 x$ and convert the rest to tangents using $\sec^2 x = 1 + \tan^2 x$. Substitute $u = \tan x$.
• If the power of tangent is odd, save a factor of $\sec x \tan x$ and convert the rest to secants using $\tan^2 x = \sec^2 x - 1$. Substitute $u = \sec x$.

3. Trigonometric Substitution

Trigonometric substitution is used to remove radicals from integrals. It is particularly useful when the integrand contains expressions of the form $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$.

Expression in Integrand	Substitution	Identity
$\sqrt{a^2 - x^2}$	$x = a \sin \theta$	$1 - \sin^2 \theta = \cos^2 \theta$
$\sqrt{a^2 + x^2}$	$x = a \tan \theta$	$1 + \tan^2 \theta = \sec^2 \theta$
$\sqrt{x^2 - a^2}$	$x = a \sec \theta$	$\sec^2 \theta - 1 = \tan^2 \theta$

After performing the substitution and integration, it is necessary to return to the original variable $x$. This is often done using a reference triangle.

4. Partial Fractions

Partial fraction decomposition is a method used to integrate rational functions (polynomial divided by polynomial). The goal is to express a complex rational function as a sum of simpler fractions that can be integrated individually.

The procedure is as follows:

1. Ensure the degree of the numerator is less than the denominator. If not, perform long division.
2. Factor the denominator completely into linear factors $(ax + b)$ or irreducible quadratic factors $(ax^2 + bx + c)$.
3. Write the rational function as a sum of partial fractions with unknown coefficients.
4. Solve for the unknown coefficients by multiplying through by the denominator and substituting convenient values for $x$ or equating coefficients.

5. Improper Integrals

An improper integral is a definite integral where either the interval of integration is infinite, or the integrand has a vertical asymptote within the interval.

Type 1: Infinite Intervals

If the integral limit is infinite, we replace the infinite limit with a variable and take the limit as that variable approaches infinity.

Type 2: Discontinuous Integrands

If $f(x)$ is discontinuous at $c$ where $a \leq c \leq b$, the integral is defined as a limit.

If the limit exists and is finite, the integral converges; otherwise, it diverges.

6. Sequences and Series

Sequences

A sequence is an ordered list of numbers $\{a_n\}$. We say that the limit of a sequence is $L$ ($\lim_{n \to \infty} a_n = L$) if the terms $a_n$ approach $L$ as $n$ becomes arbitrarily large. If a sequence has a finite limit, it is convergent; otherwise, it is divergent.

Series

A series is the sum of the terms of a sequence: $\sum_{n=1}^{\infty} a_n$. To determine if a series converges, we analyze the sequence of partial sums $S_n = \sum_{i=1}^{n} a_i$. If $\lim_{n \to \infty} S_n$ exists and is finite, the series converges.

Tests for Convergence

Determining the convergence of infinite series is a central theme of Calculus II. Below are the most common tests.

• The Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges. (Note: If the limit is 0, the test is inconclusive).
• Integral Test: If $f$ is continuous, positive, and decreasing for $x \geq 1$ and $a_n = f(n)$, then $\sum_{n=1}^{\infty} a_n$ and $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.
• Comparison Tests:
  • Direct Comparison: Compare terms to a known series.
  • Limit Comparison: If $\lim_{n \to \infty} (a_n / b_n) = c$, where $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ behave the same.
• Ratio Test: Let $L = \lim_{n \to \infty} |a_{n+1} / a_n|$.
  • If $L < 1$, the series converges absolutely.
  • If $L > 1$ (or is infinite), the series diverges.
  • If $L = 1$, the test is inconclusive.
• Root Test: Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. The conclusions are identical to the Ratio Test.
• Alternating Series Test: If a series is of the form $\sum (-1)^{n-1} b_n$ (alternating signs), where $b_{n+1} \leq b_n$ (decreasing) and $\lim b_n = 0$, then the series converges.

7. Power Series

A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x - a)^n$, where $c_n$ are coefficients and $a$ is the center.

For any power series, one of the following is true:

1. The series converges only at $x = a$.
2. The series converges for all real numbers $x$.
3. The series converges for $|x - a| < R$ (Radius of Convergence) and diverges for $|x - a| > R$. At the endpoints, tests must be performed individually.

Taylor and Maclaurin Series

A Taylor series of a function $f$ centered at $a$ is given by:

If $a = 0$, the series is called a Maclaurin series. Common functions like $e^x$, $\sin x$, and $\cos x$ have well-known Maclaurin series expansions that converge for all $x$. These series allow us to approximate functions to a high degree of accuracy using polynomials.

8. Parametric Equations and Polar Coordinates

Parametric Equations

Instead of defining $y$ as a function of $x$, we define both $x$ and $y$ in terms of a parameter $t$:

Calculus in parametric form:

• First Derivative: dy/dx = (dy/dt) / (dx/dt) provided dx/dt 0.
• Second Derivative: dy/dx = d/dt (dy/dx) / (dx/dt).
• Arc Length: $L = \int_{\alpha}^{\beta} \sqrt{(dx/dt)^2 + (dy/dt)^2} dt$.

Polar Coordinates

In the polar coordinate system, a point is determined by a distance $r$ from the origin and an angle $\theta$ from the polar axis.

Area in Polar Coordinates: The area enclosed by the curve $r = f(\theta)$ between $\theta = \alpha$ and $\theta = \beta$ is:

This section concludes the standard curriculum for Calculus II. Mastery of these integration techniques and the understanding of infinite series provides the necessary tools for solving complex engineering and physics problems in future coursework.