Mathematics · live now

Double integrals

Integration over a region of the plane: the double integral as a limit of Riemann sums, iterated integrals over rectangles, regions bounded by curves and the integrals over them in either order, reversing the order of integration, area and average value, and improper double integrals.

See pricingPreview a lesson
See pricingPreview a lesson

No account needed to read a lesson · 30-day money-back guarantee

See the full learning path

6

lessons

71

practice questions

116

prerequisite lessons

See it taught

1 of the 6 lessons here is free to preview - the whole explanation and worked example, with no account and no email.

Free preview

Double integrals over rectangles

Learn why a double integral over a rectangle can be evaluated as an iterated integral, in either order, by slicing the solid under the surface into slabs, then write the iterated integral by reading the limits of integration off the rectangle and evaluate it inside out.

∬Rf(x,y) dA=∫ab ⁣∫cdf(x,y) dy dx\iint_R f(x,y)\,dA = \int_a^b\!\int_c^d f(x,y)\,dy\,dx

Live, not a screenshot

Lesson 03 · Double integrals over rectangles

x
y
z
R
The solid between the square R and the surface z=6−x2−y22, cut into n slabs, with one picked out in red. The slabs only approximate the solid, sometimes falling short of the surface and sometimes poking through it, and the mismatch shrinks as they thin. Choose the direction of the cut and how many slabs, and drag the scene to rotate it.
Slabs6
n=6 slabs, each of thickness Δx=0.5
Slab volumes add to 43.5, against a signed volume of 43.875

This is the diagram from the lesson itself, running here. Drag it - the lesson is built round the thing it shows, not round a picture of it.

6 of 10 lessons

6 lessons you can take today, and 4 more being written. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

01

Double Riemann sums and the double integral

13 questions

Learn how a double Riemann sum estimates the volume under a surface over a rectangle, how to evaluate one from a grid of sample heights, and how the double integral is defined as the number those sums approach, including what a negative value means.

Covers

Double Riemann sums

Evaluating a double Riemann sum

Double integral notation and signed volume

02

Iterated integrals

13 questions

Learn to read the integral of a function of two variables in one variable as the area under a cross-section, evaluate it with the other variable held constant, and evaluate an iterated integral with constant limits.

Covers

The area under a cross-section

Integrating with respect to one variable

Evaluating an iterated integral

03

Double integrals over rectangles

Preview

14 questions

Learn why a double integral over a rectangle can be evaluated as an iterated integral, in either order, by slicing the solid under the surface into slabs, then write the iterated integral by reading the limits of integration off the rectangle and evaluate it inside out.

Covers

Fubini's theorem

Writing a double integral over a rectangle as an iterated integral

Evaluating a double integral over a rectangle

Preview this lesson - no account needed

04

Describing a region bounded by curves

12 questions

Learn how to describe a region bounded by curves with inequalities, first with vertical lines through it and then with horizontal ones.

Covers

Entering and leaving curves

The outer limits

Horizontal lines through RR

05

Double integrals over general regions

9 questions

Learn how to write a double integral over a region bounded by curves as an iterated integral in either order, and how to evaluate an inner integral whose limits are curves.

Covers

Writing the iterated integral

The inner integral with curves as limits

06

Evaluating a double integral over a general region

10 questions

Learn how to choose the direction of the lines through a region bounded by curves so that one iterated integral covers it, then evaluate the double integral of a function over that region all the way to a number.

Covers

Choosing the direction from the region

Evaluating the double integral

—

Reversing the order of integration

In drafting

—

Area and average value by double integration

In drafting

—

Improper double integrals

In drafting

—

What a double integral measures

In drafting

Where this unit sits in the curriculum

This unit is part of our Essential Calculus for ML learning path, which contains 154 lessons. Every one of them is drawn below.

The diagnostic test lets you skip any lesson you already know, including the ones in this unit.

Every dot is a lesson and every line a prerequisite. The 6 red dots are the lessons in Double integrals; the 116 blue dots feeding into them are the lessons Double integrals depends on; the 32 pale dots are the rest of the learning path, which come after Double integrals or alongside it.Double integralsFirst lessons on the left154 lessons

Hover any dot to name its lesson.

6 lessons in this unit

116 prerequisite lessons across the unit, counting every step back to the start

32 other lessons in the learning path - after this unit, or alongside it

Every dot is a lesson, every line a prerequisite. You can only start a lesson once you have mastered all of its prerequisites, so you are always building on solid foundations.

Builds on: Integrals, Partial derivatives

Start Double integrals

One subscription covers every learning path, and you can test out of anything you already know.

See pricingPreview a lesson
See pricingPreview a lesson

One subscription covers every learning path · 30-day money-back guarantee

More in Calculus: Limits, Derivatives, Integrals, Partial derivatives