Important Terms and Concepts

  • Definite integral
  • Riemann sum
  • Differentials
  • Integral sign
  • Integrand
  • Integral
  • Indefinite integration
  • Integral of the power function
  • Integral of the natural exponential function
  • Derivative of base-b exponential functions
  • Integral of base-b exponential functions
  • Partition
  • Subintervals
  • Sample points
  • Upper/lower bound
  • Left/right Riemann sum
  • Upper/midpoint/lower Riemann sum
  • Upper/lower limits of integration
  • Rolle's theorem
  • Mean value theorem
  • Completeness axiom
  • Fundamental theorem of calculus
  • Telescope, applied to middle terms that cancel each other out
  • Lemma
  • Differential equation
  • Limits of integration
  • Properties of definite integrals
  • Integral between symmetric limits
  • Even/odd functions
  • Disk
  • Washer
  • Solid of revolution
  • Volumes by plane slices
  • Simpson's rule

Learning Outcomes

In this chapter, students will...
  • Given the equation for a function f and a fixed point on its graph, find an equation for the linear function that best fits the given function. Use the equation to find approximate values for f(x) and values for the differentials dx in dy.
  • Become familiar with the symbol used for an indefinite integral or an antiderivative by using this symbol to evaluate indefinite integrals.
  • Use Riemann sums and limits to define and estimate values of definite integrals
  • Learn the mean value theorem and Rolle's theorem and learn how to find the point in an interval at which the instantaneous rate of change of the function equals the average rate of change.
  • Derive the fundamental theorem of calculus, and use it to evaluate definite integrals algebraically.
  • Be abel to evaluate quickly a definite integral, in an acceptable format, using the fundamental theorem of calculus.
  • Given a problem in which a quantity y varies with x, learn a systematic was to write a definite integral for the product of y and x, and evaluate the integral by using the fundamental theorem of calculus.
  • Given a solid whose cross-sectional area varies along its length, find its volume by slicing into slabs and performing appropriate calculus, and show that the answer is reasonable.
  • Use their graphing calculator's built-in numerical integration feature or Simpson's rule to find approximate values of definite integrals.

Chapter 5 Sections

5-1 Integrals of the Reciprocal Function, a Population Growth Problem
Objective: Work with the problems in this section on your own or with your study group as an assignment after completing chapter 4.
Homework: Problems 1-6 all

5-2 Linear Approximations and Differentials
Objective: Given the equation for a function f and a fixed point on its graph, find an equation for the linear function that best fits the given function. Use the equation to find approximate values for f(x) and values for the differentials dx in dy.
Homework: Q Problems 1-10, Problems 1-5 odd, 6, 7-41 odd.

5-3 Formal Definition of Antiderivative and Indefinite Integral
Objective: Become familiar with the symbol used for an indefinite integral or an antiderivative by using this symbol to evaluate indefinite integrals.
Homework - Day 1: Q Problems 1-10, Problems 1-37 odd.
Homework - Day 2: Problems 2-42 even.

5-4 Riemann Sums and the Definition of Definite Integrals
Objective: Use Riemann sums and limits to define and estimate values of definite integrals
Homework: Q Problems 1-10, Problems 1-11 odd.

5-5 The Mean Value Theorem and Rolle's Theorem
Objective: Learn the mean value theorem and Rolle's theorem and learn how to find the point in an interval at which the instantaneous rate of change of the function equals the average rate of change.
Homework - Day 1: Q Problems 1-10, Problems 1, 2, 3-21 odd.
Homework - Day 2: Problems 23-33 odd, 34, 39-41 all.

5-6 The Fundamental Theorem of Calculus
Objective: Derive the fundamental theorem of calculus, and use it to evaluate definite integrals algebraically.
Homework: Q Problems 1-10, Problems 1, 7-10 all

5-7 Definite Integral Properties and Practice
Objective: Be able to evaluate quickly a definite integral, in an acceptable format, using the fundamental theorem of calculus.
Homework - Day 1: Q Problems 1-10, Problems 1-23 odd
Homework - Day 2: Problems 25-37 odd, 38

5-8 Definite Integrals Applied to Area and Other Problems
Objective: Given a problem in which a quantity  varies with , learn a systematic way to write a definite integral for the product of  and , and evaluate the interval by using the fundamental theorem.
Homework - Day 1: Q Problems 1-10, Problems 1-11 odd
Homework - Day 2: Problems 13-31 odd

5-9 Volume of a Solid by Plane Slicing
Objective: Given a solid whose cross-sectional area varies along its length, find it’s volume by slicing into slabs and performing appropriate calculus, and show that the answer is reasonable.
Homework - Day 1: Q Problems 1-10, Problems 1-4 all
Homework - Day 2: Problems 5-9 all
Homework - Day 3: Problems 11-15 all
Homework - Day 4: Problems 16, 17-23 odd

5-10 Definite Integrals Numerically by Graphing Calculator and by Simpson’s Rule
Objective: Use your calculator’s built-in numerical integration feature or Simpson’s rule to find approximate values of definite integrals.
Homework: Q Problems 1-10, Problems 1, 4, 5, 6, 7, 11, 15, 16