This course begins with a thorough exploration of the foundational concepts of integration through Darboux sums. In the first two weeks, you will gain a solid understanding of the upper and lower Darboux sums, along with the Darboux integral itself. We will also discuss key inequalities related to these sums.
Moving forward, we delve into the necessary and sufficient conditions for Darboux integrability, laying the groundwork for an understanding of Riemann&rsquo;s definition of integrability. By weeks three to five, we will see how Riemann sums lead to a definition of integrability that is equivalent to Darboux&rsquo;s, providing a comprehensive framework for understanding these crucial concepts.
In week six, you will be introduced to the Riemann-Stieltjes integral, along with illustrative examples that will clarify its significance in mathematical analysis.
Weeks seven to nine focus on specific classes of functions, including monotone and continuous functions, to illustrate their Riemann integrability. You will also learn about the essential properties that define Riemann integrable functions.
In week ten, the course expands to include the integrability of piecewise continuous and piecewise monotone functions. We will explore the intermediate value theorem for integrals, a powerful tool in mathematical analysis.
The fundamental theorems of calculus, which form the backbone of integral calculus, will be discussed in week eleven. You will learn how these theorems connect differentiation and integration in a profound and elegant way.
Weeks twelve and thirteen offer practical tools and methods of integration, including substitution, integration by parts, and applications such as calculating volumes by slicing, cylindrical shells, and determining the length of curves and the area of surfaces of revolution.
The course concludes with a study of improper integrals in weeks fourteen and fifteen. You will explore the different types of improper integrals, their convergence, and delve into special functions such as the beta and gamma functions, examining their properties and applications.
Essential Readings: 
1. Ross, Kenneth A. (2013). Elementary Analysis: The Theory of Calculus (2nd ed.). Undergraduate Texts in Mathematics, Springer. 
2. Anton, Howard, Bivens Irl, and Davis Stephens (2012). Calculus (10th ed.). John Wiley &amp; Sons, Inc. 
3. Denlinger, Charles G. (2011). Elements of Real Analysis, Jones &amp; Bartlett India Pvt. Ltd. 
4. Ghorpade, Sudhir R., and Limaye, B. V. (2006). A Course in Calculus and Real Analysis. Undergraduate Texts in Mathematics, Springer (SIE). Indian Reprint.

Digital Products

DSC-8: Riemann Integration