# UBC MATH 101 Integral Calculus: Topics, Study Guide and Practice

> A study guide for UBC MATH 101 (Integral Calculus with Applications): definite integrals, techniques, applications and series, with practice.

Source: https://practicecalculus.com/courses/ubc-math-101
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 101 (MATH_V 101, Vancouver campus)
- **Credits:** 3
- **Prerequisites:** One of MATH 100, 102, 104, 110, 120, 180, 184 or MATH_O 100
- **Equivalency:** MATH 103, MATH 105
- **Covers:** The definite integral, integration techniques, applications, modelling and infinite series
- **Next course:** MATH 200 (Calculus III)
- **Practice here:** [Calculus 1 integrals](/learn/calculus-1/integrals/undoing-derivatives) and [Calculus 2](/learn/calculus-2)

## Key takeaways

- UBC MATH 101 is a 3-credit integral calculus course on the Vancouver campus.
- It covers the definite integral, integration techniques, applications and modelling, and infinite series.
- Any of UBC's first differential calculus courses (MATH 100, 102, 104, 110, 120, 180 or 184) meets the prerequisite.
- It is equivalent to MATH 103 and MATH 105, and it leads to MATH 200.

**MATH 101** is UBC Vancouver's second-term calculus course, titled Integral Calculus with Applications. It follows [MATH 100](/courses/ubc-math-100) or another first differential calculus course, and it takes students from the definite integral through integration techniques to infinite series. This guide maps every MATH 101 topic to practice, so you can build integration skills one problem at a time.

## What UBC MATH 101 covers

The UBC Vancouver calendar summarizes MATH 101 as the definite integral, integration techniques, applications, modelling and infinite series. It is worth 3 credits with three lecture hours per week, and it is listed as equivalent to MATH 103 and MATH 105.

In a standard sequence, that is the integral half of **Calculus 1** plus the core of **Calculus 2**. The calendar does not list individual techniques or tests, so the map below covers the topics a typical integral calculus course includes under those headings: substitution and integration by parts, applications such as area and volume, and convergence of series.

## MATH 101 topic map

| MATH 101 topic | Practice | Read |
| --- | --- | --- |
| Riemann sums and the definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums), [definite integral properties](/rules/definite-integral-properties) |
| Fundamental Theorem of Calculus | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |
| Integration techniques | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts), [partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Integration by parts](/blog/integration-by-parts), [partial fractions](/blog/partial-fractions), [trig substitution](/blog/trig-substitution) |
| Area and volume | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Area between curves](/blog/area-between-curves), [volumes of revolution](/blog/volumes-of-revolution) |
| Modelling with integrals | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Antiderivatives](/blog/antiderivatives) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Infinite series | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) | [Convergence tests](/blog/convergence-tests), [geometric series](/blog/geometric-series) |
| Testing series for convergence | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Ratio test](/rules/ratio-test), [nth-term test](/rules/nth-term-test) |

## MATH 101 and its equivalents

| Course | Title | Credits | Notes |
| --- | --- | --- | --- |
| MATH 101 | Integral Calculus with Applications | 3 | General applications |
| MATH 103 | Integral Calculus with Applications to Life Sciences | 3 | Applications to probability and dynamical systems |
| MATH 105 | Integral Calculus with Applications to Commerce and Social Sciences | 3 | Also covers partial derivatives and constrained optimization |
| MATH 121 | Honours Integral Calculus | 4 | Needs 68% in MATH 120 or 80% in a first calculus course |

After MATH 101, students continue to [MATH 200](/courses/ubc-math-200), Calculus III, or another multivariable course such as MATH 253.

## Example problems at the MATH 101 level

**1. A substitution.** Evaluate $\int_0^1 2x(x^2 + 1)^3\,dx$ with $u = x^2 + 1$, $du = 2x\,dx$:

$$\int_1^2 u^3\,du = \left[\frac{u^4}{4}\right]_1^2 = \frac{16 - 1}{4} = \frac{15}{4}$$

**2. Partial fractions.** Since $\dfrac{1}{x^2 - 1} = \dfrac{1}{2}\left(\dfrac{1}{x - 1} - \dfrac{1}{x + 1}\right)$,

$$\int \frac{dx}{x^2 - 1} = \frac{1}{2}\ln\left|\frac{x - 1}{x + 1}\right| + C$$

**3. A geometric series.**

$$\sum_{n=0}^{\infty} \frac{2}{3^n} = \frac{2}{1 - \frac{1}{3}} = 3$$

## How to do well in UBC MATH 101

- **Practice choosing a technique.** The hardest part of integration is deciding what to try. The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson trains exactly that.
- **Build series intuition early.** Start with the [geometric series formula](/rules/geometric-series-formula), then learn each test in the [convergence tests guide](/blog/convergence-tests).
- **Keep a formula sheet nearby.** The [calculus formula sheet](/formulas) lists common integrals and series.
- **Simulate exams.** Mixed, timed sets in [practice](/practice) and the plan in [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam) get you ready for midterms and the final.

> **Tip:** draw the region before every area or volume problem. A ten-second sketch tells you which curve is on top and where the limits of integration are.

## Frequently asked questions

### What is UBC MATH 101?

MATH 101, Integral Calculus with Applications, is UBC Vancouver's second-term calculus course. It covers definite integrals, integration techniques, applications and modelling, and infinite series.

### What do I need before MATH 101?

One of MATH 100, 102, 104, 110, 120, 180, 184, or MATH_O 100 from UBC Okanagan.

### Is UBC MATH 101 harder than MATH 100?

Many students find it harder. Integration requires choosing a strategy, and infinite series are a new kind of reasoning. Starting practice on integration techniques early in the term helps a lot.

### MATH 101 vs MATH 103 vs MATH 105?

The calendar lists them as equivalents. MATH 103 uses applications to the life sciences and MATH 105 uses applications to commerce and social sciences, while MATH 101 has general applications.

### What comes after MATH 101?

MATH 200, Calculus III, covers multivariable calculus. MATH 253 is another multivariable calculus option that also accepts MATH 101 as a prerequisite.