Calculus II by Dr. Linda Green

Jul 10, 2024

Lecture Notes: Calculus II by Dr. Linda Green

Introduction

  • Lecturer: Dr. Linda Green, University of North Carolina at Chapel Hill
  • Topic: Techniques from Calculus II
  • Tools: Paper and pencil recommended

Area Between Two Curves

Review from Calculus I

  • Approximate area under curve with tall skinny rectangles
  • Delta x: Small change in x values
  • Sample Point (x_i^*): Point selected within each subinterval
  • Height of Rectangle: Given by f(x_i^*)
  • Area of Rectangle: Base times height (Δx * f(x_i^*))
  • Riemann Sum: Σ(Δx * f(x_i^*))
  • Integral: Limit of Riemann sums as number of rectangles goes to infinity

Calculating Area Between Two Curves

  • Curves: y = f(x) and y = g(x)
  • Region: Between x = a and b
  • Area of Rectangle: Δx * (f(x_i^) - g(x_i^))
  • Integral: ∫(f(x) - g(x)) dx from x=a to x=b
  • Conditions: Converges if f(x) ≥ g(x) or vice versa
  • Formula: ∫(top y - bottom y) dx

Example Problem

Finding Area Between Two Parabolas

  • Curves: y = x^2 + x and y = 3 - x^2
  • Intersection Points: Solve x^2 + x = 3 - x^2
  • Area: ∫(3 - x^2 - x^2 - x) dx from bounds at points of intersection
  • Final Integral: Simplify and compute

Solids of Revolution

  • Definition: 3D objects formed by rotating region around an axis
  • Discs: Cross-sections as full circles
  • Washers: Hollow cross-sections
  • Volume Formula for Discs: ∫πR^2 dx
  • Volume Formula for Washers: ∫π(R_outer^2 - R_inner^2) dx
  • Horizontal Axis Rotation: Integrate with respect to x
  • Vertical Axis Rotation: Integrate with respect to y

Cross Sections

  • Technique: Compute volume using cross-sections
  • Formula: Integral of area of cross-section along the axis
  • Example: Region as ellipse with square cross-sections, base along x-axis

Work Calculations in Physics

Constant Force

  • Formula: Work (W) = Force (F) * Distance (d)
  • Units: Newton meters (Joules) in metric, Foot-pounds in imperial

Variable Force

  • Integral Form: ∫ F(x) dx from a to b
  • Example: Work to lift satellite, using gravitational force formula

Average Value of Functions

  • Definition: Sum of function values over an interval divided by length of the interval
  • Formula: (1/(b - a)) * ∫ f(x) dx from a to b
  • Application: Mean Value Theorem of Integrals

Volumes Using Integrals

  • Method: Divide solid into slices, integrate area of cross-section
  • Example: Base as ellipse, square cross-sections perpendicular to x-axis

Length of a Curve

  • Formula: Integral of sqrt(1 + (dy/dx)^2) dx
  • Example: Length of curve y = x^(3/2) from x=1 to x=4

Improper Integrals

Type I (Infinite Intervals)

  • Definition: Integration over an infinite interval
  • Method: ∫ from a to t as t approaches infinity
  • Example: ∫(1/x^2) dx from 1 to infinity

Type II (Discontinuous Integrands)

  • Definition: Integrand goes to infinity within the interval
  • Method: Limit of integrals over subintervals avoiding the discontinuity
  • Example: ∫(x/sqrt(x^2-1)) dx from 1 to 2

Convergence Tests for Series

Geometric Series

  • Formula: Σar^n
  • Convergence: if |r| < 1

P-Series

  • Formula: Σ(1/n^p)
  • Convergence: if p > 1

Integral Test

  • Idea: Compare series to corresponding improper integrals
  • Example: Series Σ(1/n^2)

Comparison Test

  • Idea: Compare series to a known convergent or divergent series
  • Example: Comparing to geometric or p-series

Limit Comparison Test

  • Idea: Use the limit of the ratio of terms
  • Example: Series Σ(3^n/(5^n - n^2))

Ratio Test

  • Formula: Σa_n where limit |a_(n+1)/a_n| < 1

Sequences

Definitions

  • Convergent: Sequence approaches a limit
  • Divergent: Sequence does not approach a finite limit
  • Monotonic: Sequence consistently increases or decreases
  • Bounded: Sequence stays within a fixed range

Techniques

  • Calculus Methods: L'Hopital's Rule
  • Squeeze Theorem: Bound sequence between two that converge to the same limit
  • Geometric Sequences: Test if common ratio |r| < 1

Taylor Series

Definition

  • Function: Represent as infinite sum of terms computed from the values of the function's derivatives
  • Formula: Σ(f^n(a)/n!)*(x-a)^n
  • Example: e^x, sin(x), cos(x)

Convergence

  • Radius & Interval of Convergence: Analyze using ratio or root tests
  • Error Estimates: Taylor's inequality

End of Notes on this Lecture.