# Western Calculus 1501A/B Calculus II: Topics, Guide and Practice

> A study guide for Western University Calculus 1501A/B: integration techniques, the Mean Value Theorem, Taylor series, polar curves and ODEs.

Source: https://practicecalculus.com/courses/western-university-calculus-1501
Updated: 2026-11-08

## At a glance

- **Course code:** Calculus 1501A/B
- **Course weight:** 0.50 (4 lecture hours)
- **Prerequisites:** A minimum mark of 60% in one of Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B or the former Applied Mathematics 1412A/B
- **Antirequisites:** Calculus 1301A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1414A/B and 1413
- **Covers:** Techniques of integration, the Mean Value Theorem and its consequences, series, Taylor series, parametric and polar curves, first order linear and separable differential equations
- **Next course:** Calculus 2302A/B (Intermediate Calculus I)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- Calculus 1501A/B is Western's Calculus II for the mathematical and physical sciences, a 0.50 course with 4 lecture hours.
- Western recommends it for students heading into actuarial science, applied mathematics, astronomy, mathematics, physics or statistics.
- It covers integration techniques, the Mean Value Theorem, series and Taylor series, parametric and polar curves, and first order ODEs.
- You need at least 60% in Calculus 1000 or 1500, and it leads to Calculus 2302A/B.

**Calculus 1501A/B** is Calculus II for Mathematical and Physical Sciences at Western University. The calendar says students who intend to pursue a degree in actuarial science, applied mathematics, astronomy, mathematics, physics or statistics should take it. It is a half course with four lecture hours, and it combines integration techniques, a careful look at the Mean Value Theorem, series, polar curves and first order differential equations. This guide maps each topic to free practice.

## What Western Calculus 1501A/B covers

The Western academic calendar lists the topics as: techniques of integration; the Mean Value Theorem and its consequences; series, Taylor series with applications; parametric and polar curves with applications; and first order linear and separable differential equations with applications. The prerequisite is a minimum mark of 60% in Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B or the former Applied Mathematics 1412A/B.

This is a full **Calculus 2** course. The Mean Value Theorem section is what sets it apart from Calculus 1301A/B: it is the tool used to prove many results in calculus, which matters for students heading into mathematical degrees.

## Calculus 1501A/B topic map

| Calculus 1501 topic | Practice | Read |
| --- | --- | --- |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Trig integrals and trig substitution | [Trig integrals](/learn/calculus-2/integration-techniques/trig-integrals), [trig substitution](/learn/calculus-2/integration-techniques/trig-substitution) | [Trig substitution guide](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions guide](/blog/partial-fractions) |
| Mean Value Theorem and consequences | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums), [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests) |
| Taylor series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials), [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series guide](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Parametric curves | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves), [parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | |
| Polar curves | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates), [polar area](/learn/calculus-2/power-series-parametric-polar/polar-area) | [Polar slopes](/learn/calculus-2/power-series-parametric-polar/polar-slopes) |
| Separable differential equations | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) |
| First order linear equations | [Slope fields](/learn/calculus-2/improper-integrals-and-applications/slope-fields) | [Logistic growth](/learn/calculus-2/improper-integrals-and-applications/logistic-growth) |

## Calculus 1501 vs Calculus 1301

| Course | Title | Who it is for |
| --- | --- | --- |
| Calculus 1501A/B | Calculus II for Mathematical and Physical Sciences | Math, physics, statistics, actuarial and astronomy degrees (60% entry) |
| Calculus 1301A/B | Calculus II | Students who need calculus at a less rigorous level (55% entry) |

They are antirequisites. Before this course you take [Calculus 1000A/B](/courses/western-university-calculus-1000) or 1500A/B, and after it [Calculus 2302A/B](/courses/western-university-calculus-2302).

## Example problems at the Calculus 1501 level

**1. A consequence of the MVT.** If $f'(x) = 0$ for every $x$ in an interval, then $f$ is constant there: for any $a < b$, $f(b) - f(a) = f'(c)(b - a) = 0$.

**2. A first order linear equation.** Solve $y' + y = e^{x}$. The integrating factor is $e^x$, so $(e^x y)' = e^{2x}$ and

$$y = \tfrac{1}{2}e^{x} + Ce^{-x}$$

**3. A polar slope.** For $r = \theta$, $x = \theta\cos\theta$ and $y = \theta\sin\theta$, so $\frac{dy}{dx} = \frac{\sin\theta + \theta\cos\theta}{\cos\theta - \theta\sin\theta}$.

**4. Taylor series.** $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$, so $\int_0^1 \cos(x^2)\,dx \approx 1 - \frac{1}{10} + \frac{1}{216} \approx 0.905$.

## How to do well in Calculus 1501A/B

- **Practise integration until it is quick.** Every later topic, from polar area to ODEs, ends in an integral.
- **Learn the MVT with its hypotheses.** Expect questions that ask you to use it to prove an inequality or show a function is constant.
- **Know the core series.** Geometric series, p-series and the Maclaurin series for $e^x$, $\sin x$ and $\cos x$ come up constantly.
- **Classify each ODE first.** Decide whether it is separable or first order linear before you start solving.

> **Tip:** for a first order linear equation $y' + P(x)y = Q(x)$, the integrating factor is $e^{\int P(x)\,dx}$. Multiply through and the left side becomes one derivative.

## Frequently asked questions

### What is Calculus 1501A/B at Western?

Calculus 1501A/B is Calculus II for Mathematical and Physical Sciences. The calendar lists techniques of integration; the Mean Value Theorem and its consequences; series and Taylor series with applications; parametric and polar curves with applications; and first order linear and separable differential equations with applications.

### Calculus 1501 vs Calculus 1301?

They are antirequisites. Calculus 1301A/B is for students who need a full course of calculus at a less rigorous level, and needs 55% in a first calculus course. Calculus 1501 is for mathematical and physical science degrees, needs 60%, and includes the Mean Value Theorem and its consequences.

### What do I need for Calculus 1501?

A minimum mark of 60% in one of Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, or the former Applied Mathematics 1412A/B.

### What comes after Calculus 1501?

Calculus 2302A/B, Intermediate Calculus I, which covers 3D geometry, vector functions and differential calculus of several variables, then Calculus 2303A/B for multiple and vector integrals.