---
title: "MVC.AQR.4: Multivariable and Vector Integrals (Georgia Mathematics standard)"
code: MVC.AQR.4
cite_as: Georgia Standards of Excellence, Mathematics, MVC.AQR.4
subject: math
grades:
  - high-school
language: en
canonical_url: https://georgiahomeroom.org/standards/code/MVC.AQR.4
md_url: https://georgiahomeroom.org/standards/code/MVC.AQR.4.md
alternate_language_url: https://georgiahomeroom.org/es/standards/code/MVC.AQR.4.md
source_url: https://case.georgiastandards.org/ims/case/v1p1/CFDocuments/e9dd7229-3558-4df2-85c6-57b8938f6180
source_last_changed: 2024-10-09
data_built: 2026-09-18
publisher: Georgia Homeroom, a free public-interest project of Georgia Civic Data. Not affiliated with or endorsed by the Georgia Department of Education or any school district.
up: https://georgiahomeroom.org/standards/math/high-school.md
index: https://georgiahomeroom.org/standards/index.md
---

# MVC.AQR.4: Multivariable and Vector Integrals

Interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain real-life phenomena.

- Subject: Mathematics
- Appears in: High School, https://georgiahomeroom.org/standards/math/high-school.md

## Expectations

- Evaluate and apply double and triple integrals.
  - **MVC.AQR.4.1**: Integrate functions of the form $z = f(x, y)$ or $w = f(x, y, z)$ through various techniques.
  - **MVC.AQR.4.2**: Use, evaluate, and interpret double and triple integrals in terms of volume and mass.
  - **MVC.AQR.4.3**: Represent and evaluate integrals of vector functions as double and triple integrals.
- Evaluate and interpret line and surface integrals.
  - **MVC.AQR.4.4**: Apply line and surface integral to functions representing real-world phenomena.
  - **MVC.AQR.4.5**: Solve first-order exact differential equations.
  - **MVC.AQR.4.6**: Use Green’s Theorem to evaluate line integrals in the plane; use Stokes’ Theorem to evaluate line integrals in space.
  - **MVC.AQR.4.7**: Determine whether a line integral is independent of path and use line integrals in context.
  - **MVC.AQR.4.8**: Use Gauss’ Divergence Theorem to evaluate surface integrals.
  - **MVC.AQR.4.9**: Define and apply the gradient, the divergence, and the curl in terms of integrals of vector functions.

## Georgia Milestones coverage

No Georgia Milestones blueprint cites this standard.

Source: Georgia Standards of Excellence, published by the Georgia Department of Education at case.georgiastandards.org.