UTPA STEM/CBI Courses/Calculus/Integration

Course Title: Calculus I

Lecture Topic: Integration

Instructor: Dr. Dumitru Caruntu

Institution: UTPA

Backwards Design

Course Objectives

Primary Objectives- By the next class period students will be able to:

At the end of the course students should be able to:

Understand and use the concept of function through analytical expressions and graphs
Formulate problems in terms of differential calculus
Solve problems involving differential calculus and analyze the results in terms of their physical meaning
Formulate problems in terms of integral calculus
Solve problems involving integral calculus and analyze the results in terms of their physical meaning

Sub Objectives- The objectives will require that students be able to:

Accomplishing these objectives will require students to be able to:

Represent, graph and analyze functions
Understand and use the concept of limit
Understand the definition and various interpretations of derivative
Understand and use all the differentiation rules
Use differentiation to model, graph, and analyze different applications
Understand the definition of definite integral
Understand and use the fundamental theorem of calculus
Understand and use the method of substitution
Use integration to model, graph, and analyze different engineering applications

Difficulties- Students may have difficulty:

Possible, typical instructional difficulties that I predict may occur in, and/or understanding and use of

Mathematical modeling of word problems (in particular engineering applications)
Inverse functions
Evaluate limits to infinity – direct applications are asymptotes
Chain rule of differentiation
Implicit differentiation
Curve sketching
Method of Substitution
Work and volumes
Understanding the physical meaning of the mathematical concepts

Real-World Contexts- There are many ways that students can use this material in the real-world, such as:

Real-world situation in which students would use the learning objectives include:

Mechanical design
Constructions
Medical
Aerospace
Criminal justice
Business

Model of Knowledge

Concept Map
- identify numerical methods to be used for integration
- evaluate approximately the integrals
- reduce the error of numerical integration by reducing the step size
- use the method of substitution for evaluating definite integrals
- use the method of integration by parts for evaluating definite integrals

Content Priorities
- Enduring Understanding

Graphing functions
Inverse functions
Limits at infinity
Rate of change
Derivative interpretation
Derivatives of elementary functions
Differentiation rules
Implicit differentiation
Fundamental theorem of calculus
Antiderivatives
Method of substitution

- Important to Do and Know

Function representation
Definition of limits
Definition of continuity
Relative rates
Optimization
Areas
Volumes
Work

- Worth Being Familiar with

other application of integral calculus beyond the ones already mentioned

Assessment of Learning

Formative Assessment
- In Class (groups)
  - quizzes
  - exams
- Homework (individual)
  - assigned review problems
  - assigned textbook problems
Summative Assessment
- class discussion of integration concepts and applications

Legacy Cycle

OBJECTIVE

By the next class period, students will be able to: numerically integrate use the method of substitution use the method of integration by parts The objectives will require that students be able to: know and use rectangle area formula for numerical integration know derivatives, and antiderivatives, for evaluating integrals know the fundamental theorem of calculus identify the method to be used for a given integral

THE CHALLENGE

You are the CEO of a car company and you have to decide on the car model to be manufactured next. There are two car models you have to choose from. Car market is highly competitive, and the right decision will result in billions of dollars profit. The performance criterion is the distance covered by the car in 10 seconds. The maximum acceleration provided by the engines of the two models are a1=25(1-exp(-2t)), and a2=25(t^2+2t)/(t+1)^2.

GENERATE IDEAS

Students will be asked to brainstorm on

- what do you know to solve the challenge - what else do you need to solve the challenge - how relevant is the given information to the challenge - is the information sufficient

We anticipate that students will come up with

very practical things like direct measurements or testing
using a calculator to integrate
their experience from physics and differential calculus in terms of velocity and acceleration
seeing this as an inverse problem of differentiating
numerical approach
graphing the curves

MULTIPLE PERSPECTIVES

The student is asked to explore ideas about the challenge from experts, and he should be

shown a car racing video
seeking references regarding simple methods of numerical integration
seeking references regarding more complicated methods of integration (rules, chain, substitution)

RESEARCH & REVISE

Materials to help the student exploring the challenge and amend their original ideas will be introduced. Also, lectures will be provided and they will include

the area problem (including numerical methods and Riemann sums
the distance problem
definite integral
the fundamental theorem of calculus
the substitution rule

TEST YOUR METTLE

Check students understanding by using

quizzes
structured response – clicker response
dialog

GO PUBLIC

exams
presentations to the class
projects
posters

Pre-Lesson Quiz

what is the definition of velocity?
what is the definition of acceleration
can you find the area under y=x^2 for x between 1 and 3 through geometrical approximation
would it be possible to find the total distance traveled by a car if the velocity is given? If yes, how?

Test Your Mettle Quiz

Find the total distance traveled by a car in 10 sec if its velocity is given by v=3(1+t)
Find the total distance traveled by a car in 10 sec if its velocity is given by v=3(1+t)/(1+t^2)
Find the area between y=x(ln(x)) and x axis for x between 1 and 2.