From Math 2220 Class 27 Triple Integrals Some Basic Triple - - PowerPoint PPT Presentation

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

From Math 2220 Class 27

• Dr. Allen Back
• Oct. 29, 2014

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integral Problems

Use a double (or triple) integral to find the volume bounded by the four planes z = x = 1 y = 2 x + 2y + 3z = 6.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Integrating over an elementary region D leads to triple integrals such as b

a

d(x)

c(x)

h(x,y)

g(x,y)

f (x, y, z) dz dy dx.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Integrating over an elementary region D leads to triple integrals such as b

a

d(x)

c(x)

h(x,y)

g(x,y)

f (x, y, z) dz dy dx. Please note: The inner limits of integration encode the answer to the question: For fixed x and y, what is the intersection of a line x = cst, y = cst parallel to the z-axis with D?

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Integrating over an elementary region D leads to triple integrals such as b

a

d(x)

c(x)

h(x,y)

g(x,y)

f (x, y, z) dz dy dx. Please note: The outer limits b

a

d(x)

c(x) describe 1 Which lines x = cst, y = cst intersect D. 2 Or equivalently what is the projection of D into the

xy-plane.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Integrating over an elementary region D leads to triple integrals such as b

a

d(x)

c(x)

h(x,y)

g(x,y)

f (x, y, z) dz dy dx. So on all but the easiest problems you want to

1 Sketch some version of D and perhaps indicate the lines

parallel to the axis of the inner variable of integration.

2 Sketch the projection of the region into the plane of the

two outer variables.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Setting up triple integrals (especially reliably) can be hard! One helpful principle that often comes up in the inner setup is the idea that the only places where a graph z = g(x, y) can change from being below a graph z = h(x, y) to above are at points where g(x, y) = h(x, y).

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Setup the triple integral for the volume of the region bounded by the paraboloids z = 4 − x2 − y2 and z = 1 + x2 + y2.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Setup the triple integral for the volume of the region bounded by the paraboloids z = 4 − x2 − y2 2 and z = 1 + x2 + 3y2 2 . (Note the intersection here is much more complicated than the previous example (no longer a curve lying in a plane), but the setup is not much more difficult.)

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

• xy dV

for the region between the planes x + 2y − z = 0 and y − z = 0 and above the triangle with vertices (0, 0, 0), (0, 1, 0), and (1, 0, 0).

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

The volume of the solid bounded by the parabolic cylinder x = y2, the xy-plane, and the plane x + z = 1.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Setup the triple integral of f (x, y, z) over the tetrahedron with vertices (0, 0, 0), (3, 2, 0), (0, 3, 0), and (0, 0, 2).

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Setup the triple integral of f (x, y, z) over the tetrahedron with vertices (0, 0, 0), (3, 2, 0), (0, 3, 0), and (0, 0, 2). How about changing (0, 0, 2) to (1, 1, 2)?

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Triple Integrals

Setup the triple integral of f (x, y, z) over he smaller region bounded by the cylinder x2 + y2 − 2y = 0 and the planes x − y = 0, z = 0, and z = 3.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Some Basic Triple Integal Setup Problems

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Change of Coordinates

∆A ∼ r∆r Deltaθ by geometry

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Change of Coordinates

The Polar Coordinate Transformation

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Change of Coordinates

∆A ∼

• xu

xv yu yv

• ∆u∆v

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Change of Coordinates

Polar: dA = r dr dθ. Cylindrical: dV = r dr dθ dz. Spherical: dV = ρ2 sin φ dρ dθ dφ.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Polar/Spherical/Cylindrical Problems

Problem:

• D(x2 + y2)

3 2 dx dy for D the disk x2 + y2 ≤ 1.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Polar/Spherical/Cylindrical Problems

Problem: ∞

−∞ e−x2 dx.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Polar/Spherical/Cylindrical Problems

Problem: Volume of a right circular cone of height H and largest radius R.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Polar/Spherical/Cylindrical Problems

Problem: Volume of the portion of the Earth above latitude 45◦.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Polar/Spherical/Cylindrical Problems

Problem: Volume, using spherical coordinates, of a ball of radius R with a hole of radius a (centered on a diameter) drilled out of it.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Right inverses and existence of solutions

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Left inverses and uniqueness of solutions

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Consider the map F : D∗ ⊂ R2 → D ⊂ R2 defined by F(x, y) = (x2 − y2, x + y) Label the components of image points F(x, y) as (u, v); i.e. we think of the above transformation as u = x2 − y2 v = x + y

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

To study one-to-oneness and ontoness of F, consider the algebra: u v = x2 − y2 x + y = x − y v = x + y

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Adding and subtracting the above two equations: x = 1 2

• v + u

v

• y

= 1 2

• v − u

v

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

You may take as given the fact that these formulae check showing that G(u, v) = (1 2

• v + u

v

• , 1

2

• v − u

v

• )

gives the inverse of F where everything is defined; i.e. F(G(u, v)) = (u, v) G(F(x, y) = (x, y)

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

What is the natural domain of the function G? In other words, describe the largest subset (call it U) of the uv plane on which G is defined.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Find the largest set V in the xy plane so that for all (x, y) ∈ V , F(x, y) belongs to the domain U of the function G which you found above.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Find a point (x, y) so that F(x, y) = (4, 2). More generally, briefly explain why the equation F(G(u, v)) = (u, v) shows that F : V → U is onto.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Inverses from Algebra

Note F(1, 1) = (0, 2). Can there be a different point (x, y) besides (1, 1) with F(x, y) = (0, 2)? More generally, briefly explain why the equation G(F(x, y)) = (x, y) shows that F : V → U is one-to-one.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

Two Definitions Continuity at each Point of a set S: ∀x ∈ S and ∀ǫ > 0 ∃δ > 0 so that |x − y| < δ ⇒ |f (x) − f (y)| < ǫ. Uniform Continuity on a Set S: ∀ǫ > 0 ∃δ > 0 so that ∀x, y ∈ S, |x − y| < δ ⇒ |f (x) − f (y)| < ǫ.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

An Important Theorem Bolzano Weierstrass: Every bounded sequence in Rn has a convergent subsequence. This theorem is also at the heart of the proof that continuous functions with closed and bounded domains are automatically bounded and attain their extrema.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

The idea of why Bolzano Weirrstrass holds:

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

Based on Bolzano Weierstrass, one can show that every continuous function on a closed and bounded set is uniformly continuous there.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

Suppose one is considering the integrability of a continuous function f over a rectangle of area A.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

Suppose one is considering the integrability of a continuous function f over a rectangle of area A. Consider any regular partition each of whose constituent rectangles is smaller in diameter than the δ given in the definition of uniform continuity.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Why Cont. Fcns are Integrable

Suppose one is considering the integrability of a continuous function f over a rectangle of area A. Consider any regular partition each of whose constituent rectangles is smaller in diameter than the δ given in the definition of uniform continuity. Then any two Riemann sums for this partition will differ by at most ǫA. This allows one to show that all Riemann sums settle down to a single limit as the partitions becomes sufficiently fine.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

A regular partition P of the interval [a, b] × [c, d].

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

A Riemann Sum for a bounded function f : [a, b] × [c, d] → R. SP =

n

• i=1

m

• j=1

f (cij)∆xi∆yj where cij ∈ [xi−1, xi] × [yj−1, yj]. (Regular partitions have all the xi (resp. yj) equally spaced.)

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

Theorem: If f : R → R is continuous, then f is integrable; i.e.

• R f dA exists.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

We actually compute double integrals by a sequence of 1 dimensional integrals using Fubini’s Theorem: Theorem: If f is continuous then

• R

f dA = b

a

d

c

f (x, y) dy

• dx

(or b

a

d

c

f (x, y) dy

• dx.)

At the Riemann sum level this comes down to organizing your sums over all rectangles by first adding up contributions from each row of rectangles and then adding up the row totals. But there are subtleties to this!

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

Defining double integrals for bounded regions D more general than a rectangle: We use a trick to relate this case to the rectangle one. Namely, find rectangle R containing D and define f ∗(p) = f (p) if p ∈ D

• therwise

The downside of this is that such an f ∗ is discontinuous. But for reasonable sets D, the discontinuities will be unions of a small number of “graphs of continuous functions.” (i.e. sets {(x, g(x)) : x ∈ [α, β]} or {(h(y), y) : y ∈ [γ, δ]}).

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

Particularly since continuous functions with compact domains are uniformly continuous, one can readily prove: Theorem: If a bounded function f : R ⊂ R2 → R is continuous at all points of a rectangle R except possibly for points lying a finite union of graphs of continuous functions (of one variable), then f is integrable.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

And Fubini works here too: Theorem: Suppose a bounded function f : R ⊂ R2 → R is continuous at all points of a rectangle R except possibly for points lying in a finite union of graphs of continuous functions (of one variable.) Suppose also that g(x) = d

c

f (x, y) dy exists for each x ∈ [a, b]. Then

• R

f dA = b

a

g(x) dx = b

a

d

c

f (x, y) dy

• dx.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

Another example showing Fubini is not so obviously true: Consider the (infinite to the left and down) matrix of values       . . . − 1

8

− 1

4

− 1

2

1 . . . − 1

4

− 1

2

1 . . . − 1

2

1 . . . 1 . . .       where the sum of each row is 0 while the sum of the rightmost column is 1 the next rightmost 1

2, then 1 4, . . . .

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

Another example showing Fubini is not so obviously true: Consider the (infinite to the left and down) matrix of values       . . . − 1

8

− 1

4

− 1

2

1 . . . − 1

4

− 1

2

1 . . . − 1

2

1 . . . 1 . . .       where the sum of each row is 0 while the sum of the rightmost column is 1 the next rightmost 1

2, then 1 4, . . . .

So the sum of the row sums is 0 while the sum of the column sums is 2.

From Math 2220 Class 27 V1 Double Integral Problems Triple Integrals Some Basic Triple Integal Setup Problems Change of Coordinates Polar/Sph/Cyl Problems Inverses from Algebra Why Cont. Fcns are Integrable Double Integrals

Double Integrals

If one uses the infinite partition . . . 1 2n < . . . < 1 8 < 1 4 < 1 2 < 1

• f [0, 1] in both the x and y directions on the rectangle

[0, 1] × [0, 1], it is not hard to construct a step function f whose integrals over each sub-rectangle match the infinite matrix above.