Finding Volumes

In General:

Vertical Cut: Horizontal Cut:

A =topfunction

⎛⎝⎜

⎞⎠⎟−

bottomfunction

⎛⎝⎜

⎞⎠⎟dx

a

b

∫                   a≤b

A =rightfunction

⎛⎝⎜

⎞⎠⎟−

leftfunction

⎛⎝⎜

⎞⎠⎟dy

c

d

∫                   c≤d

Find the area of the region bounded by

Bounds? In terms of y: [-2,1]

Points: (0,-2), (3,1)

Right Function?

Left Function?

Area?

x =2 + y

x =4 −y2

[(4 −y2

−2

1

∫ )−(2 + y)]dy

x + y2 =4    and    x−y=2

Volume & Definite Integrals

We used definite integrals to find areas by slicing the region and adding up the areas of the slices.

We will use definite integrals to compute volume in a similar way, by slicing the solid and adding up the volumes of the slices.

For Example………………

Blobs in SpaceVolume of a blob:

Cross sectional area at height h: A(h)

Volume =

ExampleSolid with cross sectional area A(h) = 2h at height h.Stretches from h = 2 to h = 4. Find the volume.

Volumes:

We will be given a “boundary” for the base of the shape which will be used to find a length.

We will use that length to find the area of a figure generated from the slice .

The dy or dx will be used to represent the thickness.

The volumes from the slices will be added together to get the total volume of the figure.

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square.

x2 + y2 ≤1

Bounds?

Top Function?

Bottom Function?

[-1,1]

y = 1−x2

y =− 1−x2

Length? 1−x2 − − 1−x2( )

=2 1 − x2

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square.

x2 + y2 ≤1

We use this length to find the area of the square.

Length? =2 1 − x2

Area? 2 1−x2( )

2

4 1−x2( )

4 1−x2( )dx−1

1

∫

Volume?

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square.

x2 + y2 ≤1

What does this shape look like?

4 1−x2( )dx−1

1

∫Volume?

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a circle with diameter in the plane.

x2 + y2 ≤1

Length? =2 1 − x2

Area?

π 1 − x2( )

2

π 1− x2( )

π 1− x2( )dx−1

1

∫Volume?

2 1−x2

2Radius:

Using the half circle [0,1] as the base slices perpendicular to the x-axis are isosceles right triangles.

x2 + y2 ≤1

Length? =2 1 − x2

Area?1

22 1−x2

( ) 2 1−x2( )

Volume? 2 1−x2( )dx0

1

∫

Bounds? [0,1]

Visual?

The base of the solid is the region between the curve and the interval [0,π] on the x-axis. The cross sections perpendicular to the x-axis are equilateral triangles with bases running from the x-axis to the curve.

y =2 sinx

Bounds?

Top Function?

Bottom Function?

[0,π]

y =2 sinx

y =0

Length? 2 sin x

Area of an equilateral triangle?

Area of an Equilateral Triangle?

S

S

S

S/2

S

S

Sqrt(3)*S/2

S/2

Area = (1/2)b*h

=1

2⎛⎝⎜

⎞⎠⎟S( )

3

2S

⎛

⎝⎜⎞

⎠⎟=

3

4S2

The base of the solid is the region between the curve and the interval [0,π] on the x-axis. The cross sections perpendicular to the x-axis are equilateral triangles with bases running from the x-axis to the curve.

y =2 sinx

Bounds?

Top Function?

Bottom Function?

[0,π]

y =2 sinx

y =0

Length? 2 sin x

Area of an equilateral triangle?3

4(2 sin x )2

3

4(2 sin x )2dx

0

π

∫Volume?

3

4(S)2

⎛

⎝⎜⎞

⎠⎟

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square with diagonal in the plane.

x2 + y2 ≤1

We used this length to find the area of the square whose side was in the plane….

Length? =2 1 − x2

Area with the length representing the diagonal?

Area of Square whose diagonal is in the plane?

D

SS

S2 +S2 =D2

2S2 =D2

S2 =D2

2⇒ S =

D

2

Find the volume of the solid whose bottom face is the circle and every cross section perpendicular to the x-axis is a square with diagonal in the plane.

x2 + y2 ≤1

Length of Diagonal? =2 1− x2

Length of Side?

2 1−x2

2= 2 1 − x2

2(1−x2 )Area?

Volume? 2 1−x2( )dx−1

1

∫

(S =D2)