# UT Austin M 408D: Series and Multivariable Calculus Study Guide

> A study guide for UT Austin M 408D: integration methods, parametric equations, sequences, series, partial derivatives and multiple integrals, with practice.

Source: https://practicecalculus.com/courses/ut-austin-m-408d
Updated: 2026-11-03

## At a glance

- **Course code:** M 408D
- **Hours:** 4 (three lecture hours and two discussion hours a week)
- **Prerequisites:** M 408C, 408L or 408S with at least a C-
- **Covers:** Methods of integration, parametric equations, sequences, infinite series, power series, partial derivatives and multiple integrals
- **Credit overlap:** Only one of M 403L, 408D and 408M (or 308M) may be counted
- **Sequence:** Second semester of the accelerated calculus sequence, after M 408C
- **Practice here:** [Calculus 2](/learn/calculus-2) and the first units of [Calculus 3](/learn/calculus-3)

## Key takeaways

- M 408D is the second semester of the accelerated calculus sequence at UT Austin, following M 408C.
- It covers integration methods, parametric equations, sequences and series, then an introduction to partial derivatives and multiple integrals.
- It counts for 4 hours, and only one of M 403L, M 408D and M 408M may be counted.
- Its topics span most of a standard Calculus 2 course plus the core of Calculus 3.

**M 408D** is the second half of the accelerated calculus sequence at the University of Texas at Austin. Students who finish M 408C take it to cover series and an introduction to multivariable calculus in a single semester. This guide breaks the course into its topics and links each one to free practice with instant feedback.

## What UT Austin M 408D covers

According to the UT Austin catalog, M 408D treats the theory and applications of sequences and infinite series, and introduces differential and integral calculus of several variables. The listed subjects include methods of integration, parametric equations, power series, partial derivatives and multiple integrals.

Compared with a standard sequence, that is most of **Calculus 2** (integration techniques, parametric curves, sequences and series) followed by the central ideas of **Calculus 3** (partial derivatives and double and triple integrals). On the slower track, the same material is split across M 408L and M 408M.

## M 408D topic map

| M 408D topic | Practice | Read |
| --- | --- | --- |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts), [the rule](/rules/integration-by-parts-rule) |
| Trig substitution and partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions), [trig substitution](/blog/trig-substitution) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Parametric equations | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Limits at infinity](/blog/limits-at-infinity) |
| Infinite series | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) | [Geometric series](/blog/geometric-series), [geometric series formula](/rules/geometric-series-formula) |
| Convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Power series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series), [Taylor series](/blog/taylor-series) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Multiple integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |

## M 408D vs M 408L and M 408M

| Course | Title | Prerequisite | Role |
| --- | --- | --- | --- |
| M 408C | Differential and Integral Calculus | Placement exam or M 305G | First accelerated semester |
| M 408D | Sequences, Series, and Multivariable Calculus | M 408C, 408L or 408S | Second accelerated semester |
| M 408L | Integral Calculus | M 408C, 408K or 408N | Integration, sequences and series |
| M 408M | Multivariable Calculus | M 408L or 408S | Several variables, vectors and vector calculus |

M 408D and M 408M share a credit group, so you take one or the other. See the [M 408C guide](/courses/ut-austin-m-408c) for the first semester and the [M 408M guide](/courses/ut-austin-m-408m) for the full multivariable course.

## Example problems at the M 408D level

**1. Integration by parts.** Evaluate $\int x\cos x\,dx$ with $u = x$ and $dv = \cos x\,dx$:

$$\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x + \cos x + C$$

**2. A geometric series.**

$$\sum_{n=0}^{\infty} \frac{3}{4^n} = \frac{3}{1 - \frac{1}{4}} = 4$$

**3. A double integral over a rectangle.**

$$\int_0^1\!\int_0^2 xy\,dx\,dy = \int_0^1 \left[\frac{x^2 y}{2}\right]_0^2 dy = \int_0^1 2y\,dy = 1$$

## How to do well in UT Austin M 408D

- **Treat it as two courses.** Keep separate notes for series and for multivariable calculus, and practice both from the [Calculus 2](/learn/calculus-2) and [Calculus 3](/learn/calculus-3) paths.
- **Learn when to use each convergence test.** Our guide to [convergence tests](/blog/convergence-tests) gives a decision order you can use on every problem.
- **Keep the integration formulas close.** The [calculus formula sheet](/formulas) lists the integrals and series you will reuse.
- **Study the usual errors.** The list of [common calculus mistakes](/blog/common-calculus-mistakes) covers sign slips in parts and forgotten limits of integration.
- **Practice under time limits.** Use [practice sets](/practice) with fresh numbers before each exam.

> **Tip:** most series problems in a course like M 408D come back to comparing with a geometric series or a $p$-series. Learn those two families cold first.

## Frequently asked questions

### What is M 408D at UT Austin?

M 408D, Sequences, Series, and Multivariable Calculus, is the second semester of the accelerated calculus sequence. It covers sequences and infinite series and introduces differential and integral calculus of several variables, including methods of integration, parametric equations, power series, partial derivatives and multiple integrals.

### What do I need before M 408D?

The catalog prerequisite is M 408C, M 408L or M 408S with a grade of at least C-. Most students arrive from M 408C, the first semester of the accelerated sequence.

### What is the difference between M 408D and M 408M?

M 408M, Multivariable Calculus, is a full course on functions of several variables that follows M 408L. M 408D packs series and an introduction to several variables into one semester. Only one of M 403L, M 408D and M 408M may be counted.

### Is M 408D hard?

It asks a lot because it combines two different subjects, series and multivariable calculus, in one term. Practicing convergence tests and partial derivatives a little every day works better than cramming before exams.

### How should I prepare for M 408D?

Refresh u-substitution and the fundamental theorem from M 408C, since integration techniques come first. Then practice geometric series and the basic convergence tests so the series unit feels familiar.