# Delhi University DSC-8 Riemann Integration: Syllabus and Practice

> Study guide for DSC-8 Riemann Integration in DU BSc (Hons) Mathematics, UGCF 2022: Darboux sums, the Fundamental Theorems, volumes and improper integrals.

Source: https://practicecalculus.com/courses/university-of-delhi-dsc-8
Updated: 2026-11-08

## At a glance

- **Course code:** DSC-8: Riemann Integration (Discipline Specific Core 8)
- **Programme:** B.Sc. (Hons.) Mathematics, Undergraduate Curriculum Framework (UGCF) 2022
- **Credits:** 4 credits (3 lecture + 1 tutorial)
- **Semester:** Semester III
- **Eligibility:** Class XII pass with Mathematics
- **Prerequisites:** Elementary Real Analysis and Calculus
- **Covers:** Darboux and Riemann integrals, properties of integrable functions, Fundamental Theorems of Calculus, substitution and parts, volumes, arc length, surface area, improper integrals, beta and gamma functions
- **Next course:** [DSC-11: Multivariate Calculus](/courses/university-of-delhi-dsc-11) in Semester IV
- **Practice here:** [Integrals](/learn/calculus-1/integrals/undoing-derivatives) and [applications of integrals](/learn/calculus-1/applications-of-integrals/area-between-curves) in Calculus 1

## Key takeaways

- DSC-8 Riemann Integration is a 4 credit Semester III core paper in the DU B.Sc. (Hons.) Mathematics programme under UGCF 2022.
- It builds the integral rigorously from upper and lower Darboux sums, then proves which functions are integrable.
- Unit 2 covers the intermediate value theorem for integrals and the Fundamental Theorems of Calculus I and II.
- Unit 3 is applied: substitution, parts, volumes by slicing and shells, arc length, surface area, improper integrals, and beta and gamma functions.

**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](/courses/university-of-delhi-dsc-5). This guide uses the official syllabus from the DU Department of Mathematics and links each topic to free practice.

## Course at a glance

| Item | Detail |
| --- | --- |
| Semester | III |
| Credits | 4 (3 lecture + 1 tutorial) |
| Eligibility | Class XII pass with Mathematics |
| Prerequisites | Elementary Real Analysis and Calculus |
| Hours | 45 (18 + 15 + 12) |

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

| DSC-8 topic | Practice | Read |
| --- | --- | --- |
| Riemann sums | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Upper and lower sums | [Trapezoid sums](/learn/calculus-1/integrals/trapezoid-sums) | [Riemann sums practice](/practice-problems/riemann-sums) |
| The integral as signed area | [Signed area](/learn/calculus-1/integrals/signed-area) | [Definite integral properties](/rules/definite-integral-properties) |
| Properties of the integral | [Integrals checkpoint](/learn/calculus-1/integrals/integrals-checkpoint) | [Integral linearity rules](/rules/integral-linearity-rules) |
| Intermediate value theorem for integrals | [Average value](/learn/calculus-1/applications-of-integrals/average-value) | [Mean value theorem for integrals](/theorems/mean-value-theorem-for-integrals) |
| Fundamental Theorem I | [Fundamental theorem part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Fundamental Theorem II | [Fundamental theorem part 2](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) |
| Integration by substitution | [u-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts rule](/rules/integration-by-parts-rule) |
| Volume by slicing | [Cross sections](/learn/calculus-1/applications-of-integrals/volumes-cross-sections) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Volumes of solids of revolution | [Disks](/learn/calculus-1/applications-of-integrals/volumes-disks), [washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution practice](/practice-problems/volumes-of-revolution) |
| Arc length and surface area | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Definite integrals practice](/practice-problems/definite-integrals) |
| Improper integrals, Type I | [Infinite limits of integration](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Improper integrals, Type II | [Infinite discontinuities](/learn/calculus-2/improper-integrals-and-applications/infinite-discontinuities) | [Improper integrals practice](/practice-problems/improper-integrals) |
| Convergence of improper integrals | [Comparison test](/learn/calculus-2/improper-integrals-and-applications/comparison-test) | [Integral of 1/x squared](/integrals/1-over-x-squared) |

## 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.**

$$\int_0^1 \frac{dx}{\sqrt{x}} = \lim_{a \to 0^+}\left[2\sqrt{x}\right]_a^1 = 2$$

**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](/learn/calculus-1/integrals/antiderivative-practice) and the [integration techniques](/learn/calculus-2/integration-techniques/techniques-practice) unit.
- **Study in short daily sessions** with [practice](/practice) and the [formula sheet](/formulas).

## Frequently asked questions

### What is DSC-8 at Delhi University?

DSC-8 Riemann Integration is the eighth Discipline Specific Core course in the University of Delhi B.Sc. (Hons.) Mathematics programme under UGCF 2022. It is a Semester III paper with 3 lectures and 1 tutorial a week, worth 4 credits. Its prerequisites are Elementary Real Analysis and Calculus.

### What are the units of DSC-8 Riemann Integration?

Unit 1 (18 hours) is the Riemann integral through Darboux sums and Riemann sums, plus the Riemann-Stieltjes integral. Unit 2 (15 hours) is properties of the integral and the Fundamental Theorems. Unit 3 (12 hours) is methods and applications of integration and improper integrals.

### Which book should I use for DU Riemann Integration?

The essential readings are Ross, Elementary Analysis; Anton, Bivens and Davis, Calculus; Denlinger, Elements of Real Analysis; and Ghorpade and Limaye, A Course in Calculus and Real Analysis. Bartle and Sherbert and Kumar and Kumaresan are suggested readings.

### How is DSC-8 different from school integration?

School integration focuses on finding antiderivatives. DSC-8 asks why the integral exists: you define it with Darboux sums, prove that monotone and continuous functions are integrable, and prove the Fundamental Theorems before using them.