AP CALCULUS AB ACADEMIC LIBRARY

Unit 6: Definite Integrals

1. Introduction to Unit 6
   • Instruction – Unit 6 Introduction
2. Estimating with Finite Sums
   • Instruction – Learn about estimating with finite sums.
   • Example Problem: Using Approximation Methods. How are the approximation methods used to estimate area under a curve?
   • Example Problem: Relating Distance to a Velocity vs. Time Graph. How does the area under the curve apply to velocity?
   • Example Problem: Approximating the Amount Accumulated. How do you estimate the amount accumulated?
   • Example Problem: Connecting the Area under the Curve to the Volume of a Sphere. How can you extend the area under a curve to estimate the volume of a sphere?
   • Example Problem: Estimating Population Density. How do you apply approximations to determine population density?
3. Reading Lesson 6.1: Estimating with Finite Sums
   • Instruction – Reading Lesson
   • Practice Problems 6.1 – Read section 6.1 in the textbook and complete associated exercises.
4. Definite Integrals
   • Instruction – Learn about definite integrals.
   • Example Problem: Expressing Riemann Sums as Definite Integrals. How do you use integral notation to express a limit of Riemann sums?
   • Example Problem: Using Area to Evaluate Definite Integrals. How do you evaluate definite integrals using area formulas?
   • Example Problem: Writing Accumulator Functions. How can you use definite integrals to solve problems using accumulation?
   • Example Problem: Evaluating Definite Integrals of Non-Continuous Functions. How do you evaluate definite integrals with discontinuities?
5. Reading Lesson 6.2: Definite Integrals
   • Instruction – Reading Lesson
   • Practice Problems 6.2 – Read section 6.2 in the textbook and complete associated exercises.
6. Definite Integrals and Antiderivatives
   • Instruction – Learn about definite integrals and antiderivatives.
   • Example Problem: Using the Properties of Definite Integrals. What are the properties of definite integrals?
   • Example Problem: Using the Properties of Definite Integrals. How can you calculate definite integrals using their properties?
   • Example Problem: Using the Properties of Definite Integrals to Evaluate the Upper and Lower Bounds for an Integral. How can you evaluate the upper and lower bounds of a definite integral?
   • Example Problem: Using the Mean Value Theorem. How can you use the mean value theorem for definite integrals to find the point where the average value occurs?
   • Example Problem: Calculating the Integral with Antiderivatives. How is the area under the curve related to antiderivatives?
7. Reading Lesson 6.3: Definite Integrals and Antiderivatives
   • Instruction – Reading Lesson
   • Practice Problems 6.3 – Read section 6.3 in the textbook and complete associated exercises.
8. Fundamental Theorem of Calculus, Parts 1 and 2
   • Instruction – Learn about the fundamental theorem of calculus.
   • Example Problem: Applying the First Part of the Fundamental Theorem. How do you determine derivatives of functions defined with integrals?
   • Example Problem: Analyzing Graphs with the Fundamental Theorem of Calculus. How do you analyze graphs with the fundamental theorem of calculus?
   • Example Problem: Constructing a Function with the Fundamental Theorem. How can you use the fundamental theorem of calculus to build a function?
   • Example Problem: Applying the Second Part of the Fundamental Theorem. How do you evaluate definite integrals with the fundamental theorem of calculus?
   • Example Problem: Analyzing Rates of Change in the Real World. How do you analyze rates of change with the fundamental theorem of calculus?
9. Reading Lesson 6.4: Fundamental Theorem of Calculus
   • Instruction – Reading Lesson
   • Practice Problems 6.4 – Read section 6.4 in the textbook and complete associated exercises.
10. Trapezoidal Rule
    • Instruction – Learn about the trapezoidal rule.
    • Example Problem: Approximating the Area under a Curve by Using the Trapezoidal Rule. How do we use the trapezoidal rule to approximate the area under a curve? 
    • Example Problem: Comparing the Trapezoidal Rule to Other Area Approximations. How does the trapezoidal rule compare with the LRAM, RRAM, and MRAM?
11. Reading Lesson 6.5: Trapezoidal Rule
    • Instruction – Reading Lesson
    • Practice Problems 6.5 – Read section 6.5 in the textbook and complete associated exercises.
    • Project: Analyzing Driving Data
    • Show that the area under a curve equates to the distance traveled.
12. Unit 6 AP Practice Questions
    • Unit 6 Free-Response Questions
    • Complete free-response questions for Unit 6.
13. Unit Test
    • Unit Test Answers

Unit 8: ​Applications of Definite Integrals

1. Introduction to Unit 8
   • Instruction – Unit 8 Introduction
2. Integral as Net Change
   • Instruction – Learn about integral as net change.
   • Example Problem: Calculating Displacement and Position. How do you calculate displacement and position given a particular velocity function?
   • Example Problem: Finding Total Distance Traveled. How do you calculate the total distance traveled given a particular velocity function?
   • Example Problem: Finding Net Change from Rate of Change Functions. How do you find the net change of a quantity from a rate of change function?
   • Example Problem: Finding Net Change from Graphs and Tables. How do you find the net change of a quantity from a rate of change given in graphical or tabular form?
3. Reading Lesson 8.1: Accumulation and Net Change
   • Instruction – Reading Lesson
   • Practice Problems 8.1 – Read section 8.1 in the textbook and complete associated exercises.
4. Areas in the Plane
   • Instruction – Learn about areas in the plane.
   • Example Problem: Calculating Area with dx Integration. How do you determine area by integrating with respect to x?
   • Example Problem: Calculating Area with dy Integration. How do you determine area by integrating with respect to y?
   • Example Problem: Calculating Area Using Subregions. How do you determine area by integrating using subregions?
5. Reading Lesson 8.2: Areas in the Plane
   • Instruction – Reading Lesson
   • Practice Problems 8.2 – Read section 8.2 in the textbook and complete associated exercises.
6. Volumes
   • Instruction – Learn about volumes.
   • Example Problem: Calculating Volumes with Disks. How do you determine volumes of solids using disks?
   • Example Problem: Calculating Volumes with Disks (dy). How do you determine volumes of solids by integrating with respect to y?
   • Example Problem: Calculating Volumes with Washers. How do you determine volumes of solids using washers?
   • Example Problem: Calculating Volumes with Known Cross Sections. How do you determine volumes of solids with known cross sections?
7. Reading Lesson 8.3: Volumes
   • Instruction – Reading Lesson 
   • Practice Problems 8.3 – Read section 8.3 in the textbook and complete associated exercises.
8. Applications from Science and Statistics
   • Instruction – Learn about applications of science and statistics.
   • Example Problem: Calculating the Work Done in Stretching a Spring. How much work is done in stretching a spring?
   • Example Problem: Finding the Work Done When the Force Is Nonlinear. How do you calculate the work done when the force is nonlinear?
   • Example Problem: Finding Constant Fluid Force and Fluid Pressure. How do you calculate the fluid force and pressure when the pressure is constant?
   • Example Problem: Finding Nonconstant Fluid Force and Fluid Pressure. How do you calculate a nonconstant fluid force and pressure?
   • Example Problem: Using the Definite Integral to Find the Probability of an Event.How do you find the probability of an event using the probability density function?
9. Reading Lesson 8.5: Applications from Science and Statistics
   • Instruction – Reading Lesson
   • Practice Problems 8.5 – Read section 8.5 in the textbook and complete associated exercises.
10. L’Hospital’s Rule and Other Applications
    • Instruction – Learn about L’Hospital’s rule and other applications. 
    • Example Problem: Applying L’Hospital’s Rule to Evaluate Limits. How do you apply L’Hospital’s rule to evaluate the limit of the indeterminate form 0/0?
    • Example Problem: Applying L’Hospital’s Rule to Evaluate Limits. How do you apply L’Hospital’s rule to evaluate the limit of the indeterminate form infinity/infinity?
    • Example Problem: Applying L’Hospital’s Rule to Other Indeterminate Forms. How do you apply L’Hospital’s rule to evaluate the limit of other indeterminate forms?
    • Example Problem: Comparing Growth Rates
11. Reading Lesson 9.2 and 9.3: L’Hospital’s Rule and Other Applications
    • Instruction – Reading Lesson
    • Practice Problems 9.2 and 9.3 – Read section 9.2 and 9.3 in the textbook and complete associated exercises.
    • Project: Calculating the Volume of an Object
      • Use calculus to calculate the volume of an object.
12. Unit 8 AP Practice Questions
    • Unit 8 Free-Response Questions
    • Complete free-response questions for Unit 8.
13. Unit Test
    • Unit Test Answers

Unit 10: Cumulative Exam

1. Cumulative Exam
2. ​Cumulative Exam Answers