Delhi University DSC-8 Riemann Integration: Syllabus and Practice

DSC-8: Riemann Integration is the Semester III core paper on integration in the B.Sc. (Hons.) Mathematics programme at the University of Delhi (DU), under the Undergraduate Curriculum Framework (UGCF) 2022. It carries 4 credits, with 3 lectures and 1 tutorial a week, and follows DSC-5 Calculus. This guide uses the official syllabus from the DU Department of Mathematics and links each topic to free practice.

Course at a glance

The learning objectives are to understand integration of bounded functions on a closed and bounded interval and its extension to infinite intervals or unbounded integrands, to learn properties of Riemann integrable functions and the Fundamental Theorems, and to see the practical uses of integration. The stated outcomes include applying Riemann sums to volumes and surfaces of solids of revolution, substitution and integration by parts, and convergence of improper integrals including the beta and gamma functions.

What DSC-8 covers

• Unit 1, The Riemann integral (18 hours): upper and lower Darboux sums, the Darboux integral, inequalities for Darboux sums, necessary and sufficient conditions for integrability, Riemann's definition by Riemann sums and its equivalence with Darboux's, and the Riemann-Stieltjes integral.
• Unit 2, Properties and Fundamental Theorems (15 hours): integrability of monotone and continuous functions, properties of integrable functions, piecewise continuous and piecewise monotone functions, the intermediate value theorem for integrals, and the Fundamental Theorems of Calculus I and II.
• Unit 3, Applications and improper integrals (12 hours): substitution and integration by parts, volume by slicing and cylindrical shells, arc length, area of surfaces of revolution, improper integrals of Type I, Type II and mixed type, their convergence, and the beta and gamma functions.

DSC-8 topic map

Example problems at the DSC-8 level

These are original practice problems, not past DU questions.

1. Darboux sums. For f(x) = x on [0, 1] with n equal parts, U(f, P_n) = \frac{n + 1}{2n} and L(f, P_n) = \frac{n - 1}{2n}. Their difference \frac{1}{n} \to 0, so f is integrable and \int_0^1 x\,dx = \frac{1}{2}.

2. Fundamental Theorem I. If F(x) = \int_0^{x^2}\sin t\,dt, then by the chain rule F'(x) = 2x\sin(x^2).

3. A Type II improper integral.

4. Beta function. B(2, 3) = \frac{\Gamma(2)\Gamma(3)}{\Gamma(5)} = \frac{1 \cdot 2}{24} = \frac{1}{12}, which matches \int_0^1 x(1 - x)^2\,dx.

How to do well in DSC-8

• Get comfortable with sup and inf. Darboux sums are built from them, and many proofs in Unit 1 come down to choosing a fine enough partition.
• Know which theorem you are using. Exam answers on the Fundamental Theorems lose marks when the hypotheses (continuity, integrability) are not checked.
• Keep computing skills fresh with antiderivative practice and the integration techniques unit.
• Study in short daily sessions with practice and the formula sheet.