# UW MATH 224 Multivariable Calculus: Study Guide and Practice

> A study guide for University of Washington MATH 224 (Advanced Multivariable Calculus): double and triple integrals, line and surface integrals, Stokes.

Source: https://practicecalculus.com/courses/university-of-washington-math-224
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 224
- **Credits:** 4 (one quarter)
- **Offered:** Autumn, winter, spring and summer
- **Prerequisites:** Minimum grade of 2.0 in MATH 126 or MATH 136
- **Covers:** Multivariate integration with double and triple integrals, vector fields, line and surface integrals, Green's and Stokes' theorems and the Divergence Theorem
- **Place in sequence:** Fourth quarter of the calculus sequence, after MATH 126
- **Practice here:** [Calculus 3: multiple integrals and vector calculus](/learn/calculus-3)

## Key takeaways

- MATH 224 is the fourth quarter of the UW calculus sequence, worth 4 credits and offered every quarter.
- It introduces multivariate integration: double and triple integrals, vector fields, and line and surface integrals.
- It culminates with Green's theorem, Stokes' theorem and the Divergence Theorem.
- Together with MATH 126, it covers what a semester school calls Calculus 3.

**UW MATH 224**, Advanced Multivariable Calculus, is the fourth quarter of calculus at the University of Washington. It is a 4-credit course offered every quarter that finishes multivariable calculus with integration and vector calculus. This guide maps every MATH 224 topic to practice with instant feedback.

## What UW MATH 224 covers

The UW course catalog describes MATH 224 as an introduction to multivariate integration, including double and triple integrals, vector fields, and line and surface integrals. It culminates with the theorems of Green and Stokes and the Divergence Theorem. It requires a 2.0 in MATH 126 or MATH 136.

At a semester school, this is the second half of **Calculus 3**. Our Calculus 3 path covers it in two units: multiple integrals, then vector calculus.

## MATH 224 topic map

| MATH 224 topic | Practice | Read |
| --- | --- | --- |
| Double integrals over rectangles | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Changing the order of integration | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Double integrals](/blog/double-integrals) |
| General regions | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Polar coordinates | [Double integrals in polar coordinates](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Area in polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-area) |
| Applications: area, volume, mass | [Area, volume, mass and average value](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Double integrals](/blog/double-integrals) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Change of variables | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [Double integrals](/blog/double-integrals) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Line integrals | [Work and line integrals](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Conservative fields | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Green's theorem](/theorems/greens-theorem) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Curl, divergence and surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Divergence theorem](/theorems/divergence-theorem) |
| Stokes' theorem and the Divergence Theorem | [Stokes and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## How MATH 224 fits with MATH 126

| | MATH 126 | MATH 224 |
| --- | --- | --- |
| Title | Calculus with Analytic Geometry III | Advanced Multivariable Calculus |
| Credits | 5 | 4 |
| Multivariable focus | Vectors, vector functions, partial derivatives | Multiple integrals, vector calculus |
| Also covers | Taylor polynomials and series | Green's, Stokes' and Divergence theorems |

Students in the honors track reach MATH 224 through MATH 136 instead of MATH 126. See the [MATH 126 guide](/courses/university-of-washington-math-126) for the prerequisite material and the [MATH 125 guide](/courses/university-of-washington-math-125) for the integration skills you will reuse.

## Example problems at the MATH 224 level

**1. Switching the order.** The integral $\int_0^1 \int_x^1 e^{y^2}\,dy\,dx$ cannot be done as written. The region is $0 \le x \le y \le 1$, so

$$\int_0^1 \int_0^y e^{y^2}\,dx\,dy = \int_0^1 y e^{y^2}\,dy = \frac{e - 1}{2}$$

**2. A spherical volume.** The volume of a ball of radius $a$ is

$$\int_0^{2\pi}\int_0^{\pi}\int_0^a \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta = 2\pi \cdot 2 \cdot \frac{a^3}{3} = \frac{4}{3}\pi a^3$$

**3. The Divergence Theorem.** For $\mathbf{F} = \langle x, y, z \rangle$ over the unit ball, $\nabla \cdot \mathbf{F} = 3$, so the outward flux is

$$\iiint_E 3\,dV = 3 \cdot \frac{4}{3}\pi = 4\pi$$

## How to do well in UW MATH 224

- **Sketch before you integrate.** Bounds come from the picture. Practicing [general regions](/learn/calculus-3/multiple-integrals/general-regions) builds the habit.
- **Pick coordinates that fit the region.** Circles suggest polar, cylinders cylindrical, spheres spherical. The [formula sheet](/formulas) lists each volume element.
- **Learn the theorems as a family.** Start with [Green's theorem](/theorems/greens-theorem), then read [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence theorem](/theorems/divergence-theorem) as higher-dimensional versions.
- **Test yourself** with timed [practice exams](/practice) and review [common calculus mistakes](/blog/common-calculus-mistakes).

> **Tip:** if a surface integral looks painful and the surface is closed, try the Divergence Theorem. A triple integral of the divergence is often much easier.

## Frequently asked questions

### What is MATH 224 at the University of Washington?

MATH 224, Advanced Multivariable Calculus, is the fourth quarter of UW's calculus sequence. It introduces double and triple integrals, vector fields, and line and surface integrals, and it culminates in the theorems of Green and Stokes and the Divergence Theorem.

### What do I need for MATH 224?

A minimum grade of 2.0 in MATH 126 or in MATH 136, the honors equivalent.

### Is MATH 224 the same as Calculus 3?

It is the integration half of a standard Calculus 3 course. The differentiation half, with vectors and partial derivatives, is in MATH 126.

### Is MATH 224 hard?

The main challenge is setting up integrals: choosing coordinates, finding bounds and parametrizing curves and surfaces. Once the setup is right, the computations use familiar integration skills.

### How should I prepare for MATH 224?

Review partial derivatives and the gradient from MATH 126, polar coordinates, and single-variable integration techniques. Being fluent with the dot and cross product also helps with line and surface integrals.