# UTPA STEM/CBI Courses/Calculus/Integration

Course Title: Calculus I

Lecture Topic: Integration

Instructor: Dr. Dumitru Caruntu

Institution: UTPA

## Backwards Design[edit | edit source]

**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

- In Class (groups)
**Summative Assessment**- class discussion of integration concepts and applications

## Legacy Cycle[edit | edit source]

**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[edit | edit source]

- 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[edit | edit source]

- 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.