Calculus Study Guide: Techniques of Integration, Applications, Series, and Partial Differentiation

Techniques of Integration

Simple Techniques and Trigonometric Identities

Integration often requires transforming the integrand into a more manageable form. Trigonometric identities are especially useful for simplifying products and powers of trigonometric functions.

• Key Identities:

  • cos 2A = 2 cos^2 A − 1

  • sin^2 A = (1 − cos 2A)/2

  • sin A cos B = [sin(A − B) + sin(A + B)]/2

  • cos A cos B = [cos(A − B) + cos(A + B)]/2

  • sin A sin B = [cos(A − B) − cos(A + B)]/2

• Example: To integrate $\int \sin 3x \cos 2x\,dx$, use the product-to-sum identity.

Differentials

The differential dy of a function y = f(x) is defined as $dy = f'(x)dx$. This concept is foundational for substitution in integration and for understanding the chain rule.

• Example: If $y = \sqrt{1 + x}$, then $dy = \frac{1}{2\sqrt{1 + x}}dx$.

Integration by Substitution

Substitution is used when the integrand contains a function and its derivative. Let $u = g(x)$, then $du = g'(x)dx$ and $\int f(g(x))g'(x)dx = \int f(u)du$.

• Example: $\int \sin^3 x \cos x dx$ with $u = \sin x$, $du = \cos x dx$.

Indirect (Trigonometric) Substitution

For integrals involving square roots of quadratic expressions, trigonometric substitutions are effective. The choice of substitution depends on the form:

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

Partial Fractions

Rational functions can often be decomposed into simpler fractions, which are easier to integrate. This is especially useful when the degree of the numerator is less than the denominator.

• Example: $\int \frac{dx}{x^2 + x - 2}$ can be split into partial fractions.

Integration by Parts

This technique is based on the product rule for differentiation. For $\int u dv = uv - \int v du$.

• Guideline: Choose u and dv such that differentiating u and integrating dv simplifies the integral. The LIATE rule (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) helps prioritize choices.

• Example: $\int x e^x dx$ with $u = x$, $dv = e^x dx$.

Applications of Definite Integrals

Areas and Arc Lengths

Definite integrals are used to compute areas under curves, between curves, and arc lengths. For a curve $y = f(x)$ from $x = a$ to $x = b$:

• Area: $A = \int_a^b |f(x)| dx$

• Arc Length: $L = \int_a^b \sqrt{1 + (f'(x))^2} dx$

Parametric and Polar Curves

For curves given by $x = x(t)$, $y = y(t)$, the area and arc length are:

• Area: $A = \int_{t_1}^{t_2} y(t) x'(t) dt$

• Arc Length: $L = \int_{t_1}^{t_2} \sqrt{(x'(t))^2 + (y'(t))^2} dt$

For polar curves $r = r(\theta)$:

• Area: $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$

• Arc Length: $L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta$

Areas Enclosed by Loops

For curves with loops, the area can be found by integrating over the parameter values that trace the loop.

Curved Surface Area

When a curve $y = f(x)$ is revolved about the x-axis, the surface area is:

• Surface Area: $S = 2\pi \int_a^b y \sqrt{1 + (f'(x))^2} dx$

Advanced Applications of Differentiation

Taylor Polynomials and Series

Taylor polynomials approximate functions near a point. The nth Taylor polynomial for $f(x)$ about $x = a$ is:

• $P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$

The Maclaurin series is the Taylor series about $x = 0$.

• Example: For $f(x) = e^x$, the Maclaurin series is $\sum_{n=0}^\infty \frac{x^n}{n!}$.

Taylor's Theorem and Error

The error in approximating $f(x)$ by its nth Taylor polynomial is given by:

• $f(x) - P_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$.

Binomial Series

The binomial series generalizes the binomial theorem for any real exponent $\alpha$:

• $(1 + x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \cdots$

Indeterminate Forms and L'Hôpital's Rule

For limits of the form $0/0$ or $\infty/\infty$, L'Hôpital's Rule states:

• If $\lim_{x\to a} f(x) = 0$ and $\lim_{x\to a} g(x) = 0$, then $\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).

Partial Differentiation and Quadric Surfaces

Quadric Surfaces

Quadric surfaces are second-degree surfaces in three variables. The general form is $z = Ax^2 + Bxy + Cy^2 + Dx + Ey + F$. By rotating axes and shifting the origin, canonical forms are obtained:

Form	Equation	Name
Both positive	$z = u^2/a^2 + v^2/b^2$	Elliptic paraboloid (cup)
Both negative	$z = -u^2/a^2 - v^2/b^2$	Elliptic paraboloid (cap)
One positive, one negative	$z = u^2/a^2 - v^2/b^2$	Hyperbolic paraboloid (saddle)
One zero	$z = u^2/a^2$ or $z = -u^2/a^2$	Parabolic cylinder

Partial Derivatives

For $z = f(x, y)$, the partial derivatives are:

• $\frac{\partial z}{\partial x}$: Differentiate with respect to $x$, treating $y$ as constant.

• $\frac{\partial z}{\partial y}$: Differentiate with respect to $y$, treating $x$ as constant.

Second partial derivatives include $z_{xx}$, $z_{yy}$, $z_{xy}$, and $z_{yx}$. For most well-behaved functions, $z_{xy} = z_{yx}$ (Clairaut's theorem).

Chain Rule for Partial Derivatives

If $z = f(x, y)$, and $x = x(u, v)$, $y = y(u, v)$, then:

• $\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u}$

Differentials and First Approximations

The differential $dz$ gives the first-order approximation to the change in $z$ due to small changes in $x$ and $y$:

• $dz = z_x dx + z_y dy$

Inverse Functions and Jacobians

If $x$ and $y$ are functions of $u$ and $v$, and vice versa, the matrices of partial derivatives are inverses of each other. This is important in change of variables and transformations.

Differential Equations

Types of First-Order Differential Equations

• Variable Separable: $\frac{dy}{dx} = f(x)g(y)$, solved by separating variables and integrating both sides.

• Homogeneous: $M(x, y)dx + N(x, y)dy = 0$ where $M$ and $N$ are homogeneous functions of the same degree. Use substitution $y = vx$.

• Exact: $M(x, y)dx + N(x, y)dy = 0$ is exact if $M_y = N_x$. Find a potential function $f(x, y)$ such that $df = M dx + N dy$.

• Linear: $\frac{dy}{dx} + P(x)y = Q(x)$. Use integrating factor $\mu(x) = e^{\int P(x)dx}$.

Appendix: Common Curve Sketches

Additional info: This guide covers the main topics from the provided Calculus course material, including techniques of integration, applications of definite integrals, Taylor and Maclaurin series, partial differentiation, and first-order differential equations. For each topic, key definitions, formulas, and examples are provided, with relevant images included to reinforce understanding where appropriate.