Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a...

Triple IntegralsMath 212

Brian D. Fitzpatrick

Duke University

February 26, 2020

Overview

Triple IntegralsMotivationIterated IntegralsNonrectangular Regions

Triple IntegralsMotivation

Example

An object conforms to the shape of a solid W in R3.

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

Suppose f ∈ C (R3) measures density (kg/m3) throughout W .

DefinitionThe triple integral of f on W is∫∫∫

Wf dV = mass of W (in kg)

Example

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

Example

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

Example

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

Example

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

Example

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

Example

f (P1)=14 kg/m3

P2f (P2)=7 kg/m3

f (P3)=9 kg/m3

P4 f (P4)=21 kg/m3

ObservationTracking units allows us to interpret double integrals.∫∫∫

mass units

volume unit

mass of W

mass units

volume unit

mass of W

mass units

volume unit

mass of W

mass units

volume unit

= mass of W

Triple IntegralsIterated Integrals

QuestionHow can we calculate a triple integral?

AnswerUse iterated integrals!

QuestionHow can we calculate a triple integral?

AnswerUse iterated integrals!

Example

Suppose f = xy − z ◦C/m3 measures density throughout

W = [1, 3]× [3, 5]× [−2, 2]

The mass of W is∫∫∫W

f dV =

−2xy − z dz dy dx =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx = 16 x2

∣∣x=3

34 xy dy dx = 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx = 16 x2

∣∣x=3

34 xy dy dx = 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

−2xy − z dz dy dx

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx = 16 x2

∣∣x=3

34 xy dy dx = 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx = 16 x2

∣∣x=3

34 xy dy dx = 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx

= 16 x2∣∣x=3

34 xy dy dx = 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx

= 16 x2∣∣x=3

34 xy dy dx

= 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx

= 16 x2∣∣x=3

34 xy dy dx

= 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx

= 16 x2∣∣x=3

34 xy dy dx

= 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx =

16 x2∣∣x=3

34 xy dy dx

= 128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx = 16 x2

∣∣x=3

34 xy dy dx =

128 ◦C

12 xy2

∣∣y=5

Example

W = [1, 3]× [3, 5]× [−2, 2]

f dV =

132 x dx

3xyz − 1

2z2∣∣∣∣z=2

z=−2dy dx = 16 x2

∣∣x=3

34 xy dy dx = 128 ◦C

12 xy2

∣∣y=5

ObservationDouble integrals over rectangular regions come in two flavors

∫ x2

∫ y2

f dy dx

(x-slicing)

∫ y2

∫ x2

f dx dy

(y -slicing)

Triple integrals over rectangular regions come in six flavors∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

ObservationDouble integrals over rectangular regions come in two flavors∫ x2

∫ y2

f dy dx

(x-slicing)

∫ y2

∫ x2

f dx dy

(y -slicing)

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx

(x-slicing)

∫ y2

∫ x2

f dx dy

(y -slicing)

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

f dx dy

(y -slicing)

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

f dx dy

(y -slicing)

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

f dx dy (y -slicing)

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

Triple integrals over rectangular regions come in six flavors

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx

(x-slicing)

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

f dy dz dx (x-slicing)∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy

(y -slicing)

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

f dx dz dy (y -slicing)∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz

(z-slicing)

∫ y2

f dy dx (x-slicing)

∫ y2

∫ x2

∫ y2

∫ z2

f dz dy dx

∫ x2

∫ z2

∫ y2

∫ x2

∫ z2

f dz dx dy

∫ y2

∫ z2

∫ x2

∫ y2

f dy dx dz

∫ z2

∫ y2

∫ x2

f dx dy dz (z-slicing)

Triple IntegralsNonrectangular Regions

QuestionHow do we compute

∫∫∫W f dV if W is not rectangular?

AnswerOur slicing method will depend on the shape of W .

QuestionHow do we compute

∫∫∫W f dV if W is not rectangular?

AnswerOur slicing method will depend on the shape of W .

Example

Consider the “first octant” part of x + 2 y + 3 z ≤ 6.

6−x2

6−x3

x + 2 y + 3 z = 6

Each x-slice leaves an imprint on the yz-plane.

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6−x2

6−x3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−x2

∫ 6−x−2 y3

0f dz dy dx =

∫ 6−x3

∫ 6−x−3 z2

0f dy dz dx

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

Each y -slice leaves an imprint on the xz-plane.

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 2 y

6−2 y3

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−2 y

∫ 6−x−2 y3

0f dz dx dy =

∫ 6−2 y3

∫ 6−2 y−3 z

0f dx dz dy

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

Each z-slice leaves an imprint on the xy -plane.

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

6− 3 z

6−3 z2

x + 2 y + 3 z = 6

∫∫∫W

f dV =

∫ 6−3 z

∫ 6−x−3 z2

0f dy dx dz =

∫ 6−3 z2

∫ 6−2 y−3 z

0f dx dy dz

Example

Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.

√1− x2

z = x2 + y2

∫∫∫W

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

∫∫∫W

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

∫∫∫W

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

∫∫∫W

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

∫∫∫W

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

Each x-slice leaves an imprint on the yz-plane.∫∫∫W

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

√1− x2

z = x2 + y2

f dV =

∫ √1−x20

f dz dy dx =

∫ √z−x20

f dy dz dx

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

z = x2 + y2

∫∫∫W

f dV =

∫ √z0

∫ √z−x20

f dy dx dz =

∫ √z0

∫ √z−y2

0f dx dy dz

Example

Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

Each z-slice leaves an imprint on the xy -plane.∫∫∫W

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

∫∫∫W

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

∫∫∫W

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Example

y = x2 + z2 + 4

y = 2 x2 + 2 z2

x2 + z2 = 4

y = x2 + z2 + 4

y = 2 x2 + 2 z2

√4− z2−

√4− z2

f dV =

∫ √4−z2−√4−z2

∫ x2+z2+4

2 x2+2 z2f dy dx dz

Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a...

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