**What are you going to learn?**

**Content**

Evaluate indefinite and definite integrals, involving logarithmic and exponential functions. Solve problems involving Mean – Value Theorem and the Fundamental Theorem of Calculus. Evaluate integrals by different methods of integration. Differentiate between different types of indeterminate forms and finding limit of functions. Test improper integrals for convergence. Calculate areas of plane regions and arc length. Calculate volumes by both washers and cylindrical shells methods. Calculate areas of plane regions and arc length using polar coordinates.

**Chapter 1. Definite Integrals**

Introduction

Area Under the Curve

Sigma Notation

The Definite Integral

The Mean Value Theorem

The Fundamental Theorem of Calculus

Area of a Region in a Plane

Volumen of a Solid of Revolution

Work

Liquid Pressure

Center of Mass

Length of Arc of a Plane Curve

Anti-derivatives

**Chapter 2. Applications of Definite Integral**

**Chapter 3. Integrals and Transcendental Functions**

The Logarithm Defined as an Integral

Exponential Functions

Hyperbolic Functions

**Chapter 4. techniques of Integration**

Basic Formulas

Integration by Parts

Trigonometric Integrals

Trigonometric Substitutions

Integration of Rational Functions by Partial Fractions

Numerical Integration with Python

**Chapter 5. Improper Integrals**

Infinite Limits of Integration

Integrands with Vertical Asymptotes

Improper Integrals with Python

Tests for Convergence and Divergence

Preventive Actions

**Chapter 6. First Order Differential Equations**

Solutions, Slopes, and Euler's Method

First Order Linear Equations

Applications

Systems of Equations and Phase Planes

**Chapter 7. Infinite Sequences and Series**

Introduction

Convergence tests

Alternating series

**Chapter 8. Power Series**

Sequences

Infinite Series

Integral Test

Comparison Test

Absolute Convergence: The Ratio and Root Tests

Alternating Series and Conditional Convergence

The Binomial Series

### Bibliography

Thomas, G. et al (2005).

*Thomas' Calculus:Early Transcendentals**.*11th Edition: Pearson Education, Limited.Stewart, J.

. 8th edition. Belmont, CA: Brooks Cole.*Calculus Early Transcendentals*Apostol, T. (1969).

. 2D ED. John Wiley and Sons.*Calculus Vol. 1*Spivak, M. (2006).

4th Ed, Cambridge University Press.*Calculus*.