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.

University of Washington Updated November 3, 2026 3 min read

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	Double integrals
Changing the order of integration	Switching the order	Double integrals
General regions	Double integrals over general regions	Double integrals
Polar coordinates	Double integrals in polar coordinates	Area in polar coordinates
Applications: area, volume, mass	Area, volume, mass and average value	Double integrals
Triple integrals	Triple integrals	Double integrals
Cylindrical and spherical coordinates	Cylindrical and spherical coordinates	Double integrals
Change of variables	Change of variables and the Jacobian	Double integrals
Vector fields	Vector fields	Gradient and directional derivatives
Line integrals	Work and line integrals	Dot and cross product
Conservative fields	Conservative fields	Green's theorem
Green's theorem	Green's theorem	Green's theorem
Curl, divergence and surface integrals	Surface integrals and flux	Divergence theorem
Stokes' theorem and the Divergence Theorem	Stokes and the Divergence Theorem	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 for the prerequisite material and the MATH 125 guide for the integration skills you will reuse.

Example problems at the MATH 224 level

1. Switching the order. The integral $inline$ cannot be done as written. The region is $inline$ , so

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

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3. The Divergence Theorem. For $inline$ over the unit ball, $inline$ , so the outward flux is

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How to do well in UW MATH 224

Sketch before you integrate. Bounds come from the picture. Practicing general regions builds the habit.
Pick coordinates that fit the region. Circles suggest polar, cylinders cylindrical, spheres spherical. The formula sheet lists each volume element.
Learn the theorems as a family. Start with Green's theorem, then read Stokes' theorem and the Divergence theorem as higher-dimensional versions.
Test yourself with timed practice exams and review 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.