Introduction to Calculus and Analytic Geometry II

Area Between Curves

To find the area between two curves, we integrate the difference between the upper and lower functions:

When the curves intersect multiple times or when functions are expressed in terms of y rather than x, we partition the region appropriately and sum the individual areas.

Volumes by Slicing

Volumes of solids with known cross-sectional areas can be found by integrating these areas:

Special cases include the disk method and the washer method for solids of revolution:

• Disk method: V = [a,b] [f(x)] dx
• Washer method: V = [a,b] ([f(x)] - [g(x)]) dx

Volumes by Cylindrical Shells

For solids of revolution, an alternative approach uses cylindrical shells:

This method is often simpler when rotating around a vertical axis or when the disk/washer method would require complex equations.

Arc Length

The length of a curve y = f(x) from x = a to x = b is:

This formula derives from approximating the curve by tiny line segments and taking the limit as the segment length approaches zero.

Physical Applications

Integration solves numerous physical problems:

• Work: W = [a,b] F(x) dx, where F(x) is the force
• Center of mass: x = (xdm)/(dm)
• Moments: My = x(x) dx (about the y-axis)
• Hydrostatic force: F = [a,b] gh(y)L(y) dy

Sequences

A sequence {a} is an ordered list of numbers indexed by positive integers. Key concepts include:

Monotonic (always increasing or decreasing) and bounded sequences always converge by the Monotone Convergence Theorem.

Series

A series a is the sum of the terms of a sequence. The partial sum S = [i=1 to n] a forms a new sequence. The series converges if the sequence of partial sums converges.

Tests for Convergence:

Power Series

A power series centered at a has the form:

Each power series has a radius of convergence R (possibly 0 or ) such that the series converges for |x - a| < R and diverges for |x - a| > R. The convergence at the endpoints x = a R must be checked separately.

Taylor and Maclaurin Series

Power series can represent analytic functions near a point. The Taylor series of f at a is:

When a = 0, this is called a Maclaurin series. Important Maclaurin series include:

• e^x = [n=0 to ] x/n!
• sin(x) = [n=0 to ] (-1)x/(2n+1)!
• cos(x) = [n=0 to ] (-1)x/(2n)!
• ln(1+x) = [n=1 to ] (-1)x/n

Parametric Equations

A parametric curve is defined by expressing x and y as functions of a parameter t:

Parametrics are particularly useful for describing motion and curves that don't pass the vertical line test.

Calculus with Parametrics:

• First derivative: dy/dx = (dy/dt)/(dx/dt)
• Second derivative: dy/dx = (d/dt(dy/dx))/(dx/dt)
• Arc length: L = [a,b] [(dx/dt) + (dy/dt)] dt
• Area: A = [a,b] y(dx/dt) dt

Polar Coordinates

In polar coordinates, a point is specified by its distance r from the origin and angle from the positive x-axis:

Polar coordinates excel at describing curves with circular or spiral patterns.

Calculus in Polar Coordinates:

• Area: A = [,] r d
• Arc length: L = [,] [r + (dr/d)] d
• Slope of tangent line: dy/dx = (rcos + dr/dsin)/(-rsin + dr/dcos)

Common Polar Curves

Several classic curves have elegant polar forms:

• Cardioid: r = a(1 sin) or r = a(1 cos)
• Limaon: r = a bsin or r = a bcos
• Rose: r = asin(n) or r = acos(n)
• Lemniscate: r = acos(2) or r = asin(2)
• Spiral of Archimedes: r = a

Three-Dimensional Coordinate System

Points in space are represented by ordered triples (x, y, z). The coordinate planes (xy, yz, xz) divide space into eight octants. Distance between points and equations of planes and lines all have three-dimensional analogues.

Vectors in Space

A vector has both magnitude and direction. In component form, a vector from point A to point B is:

Vector Operations:

• Addition: a + b = <a+b, a+b, a+b>
• Scalar multiplication: ca = <ca, ca, ca>
• Magnitude: |a| = (a + a + a)
• Unit vector: a/|a|

Dot and Cross Products:

Lines and Planes in Space

The mathematical descriptions of lines and planes in 3D rely on vectors:

Lines:

Planes:

Cylindrical and Spherical Coordinates

Beyond Cartesian coordinates, we have alternative 3D coordinate systems:

Cylindrical Coordinates:

Spherical Coordinates:

Functions of Several Variables

A function of two variables z = f(x,y) assigns a single output to each ordered pair (x,y) in its domain. The graph of such a function is a surface in three-dimensional space.

For functions of three variables u = f(x,y,z), level surfaces f(x,y,z) = k replace level curves.

Limits and Continuity

The definition of a limit extends to multiple variables:

A function is continuous at (a,b) if lim[(x,y)(a,b)] f(x,y) = f(a,b). The existence of limits in multiple variables is more subtle, as the function must approach the same value along all possible paths.

Partial Derivatives

To analyze functions of multiple variables, we consider derivatives with respect to each variable separately:

Higher-order partial derivatives include mixed partials like f/xy, which under suitable regularity conditions are equal (Clairaut's Theorem: f/xy = f/yx).

Tangent Planes and Linear Approximation

The tangent plane to the surface z = f(x,y) at point (a,b,f(a,b)) is:

This plane provides the best linear approximation to the function near the point of tangency. The differential dz = f_x(x,y)dx + f_y(x,y)dy approximates the change in the function.

The Gradient and Directional Derivatives

The gradient vector f(x,y) = <f_x, f_y> points in the direction of steepest increase of the function and is perpendicular to level curves.

The maximum rate of change occurs in the direction of f and equals |f|, while the minimum rate equals -|f| and occurs in the opposite direction.

Extrema of Functions of Two Variables

Critical points occur where f = 0 or f doesn't exist. The Second Derivatives Test classifies critical points:

On a closed, bounded region, the Extreme Value Theorem guarantees the existence of absolute maximum and minimum values, which occur either at critical points or on the boundary.