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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.

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118

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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.

\iint_R f(x,y)\,dA = \int_a^b\!\int_c^d f(x,y)\,dy\,dx

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What a double integral measures

Learn how to read what an integral measures, whether a length, an area, a signed volume or a total from a density, from its piece and its integrand.

\iint_R f(x,y)\,dA = \int_a^b A(x)\,dx

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Lesson 03 · Double integrals over rectangles

0,0

x

y

z

az = 0.79

el = 0.66

bank = 0.00

R

The solid between the square

R

and the surface

z = 6 - x^{2} - \frac{y^{2}}{2}

, 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.

Slabs

6

n = 6

slabs, each of thickness

Δ x = 0.5

Slab volumes add to

43.5

, against a signed volume of

43.875

0,0

x

y

z

az = 0.79

el = 0.66

bank = 0.00

R

Δ x

A (x_{k})

Cutting across

x

, the slab at

x_{k}

has an end face of area

A (x_{k})

, the cross-section there, and thickness

Δ x

, so its volume is

A (x_{k}) Δ x

. That only approximates the piece of the solid it replaces, since the surface rises or falls a little across the thickness, and it improves as the slabs thin.

Thickness

Δ x = 0.5

n = 6

Δ x = 0.5

, slab volume

A (x_{k}) Δ x = 8.063

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The 10 lessons

The 10 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

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

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

Double integrals over rectangles

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14 questions

Covers

Fubini's theorem

Writing a double integral over a rectangle as an iterated integral

Evaluating a double integral over a rectangle

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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 $R$

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

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

8 questions

Learn how to recover the region of integration from the limits of an iterated integral, describe it with lines in the other direction, and write the integral with the order of integration reversed.

Covers

Recovering the region from an iterated integral

Writing the reversed integral

Area and average value by double integration

13 questions

Learn how to find the area of a region bounded by curves as a double integral, and how to find the average value of a function of two variables over a region by dividing its double integral by the area of the region.

Covers

Area of a region as a double integral

Average value of a function over a region

Average value over a region bounded by curves

Improper double integrals

16 questions

Learn how to write a double integral over an unbounded region as an iterated integral with an infinite limit of integration, and how to evaluate it as a limit when the infinite limit sits on the inner integral, on the outer integral, or on both.

Covers

Writing an improper double integral

Evaluating with one infinite limit of integration

Evaluating with infinite limits on both integrals

What a double integral measures

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10 questions

Learn how to read what an integral measures, whether a length, an area, a signed volume or a total from a density, from its piece and its integrand.

Covers

Reading what an integral measures

Reading a density integral

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Where this unit sits in the curriculum

This unit is part of our Essential Calculus for ML learning path, which contains 191 lessons. Every one of them is drawn below.

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10 lessons in this unit

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

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Builds on: Graphs of functions, Integrals, Partial derivatives

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