Transcript Document

Finding Volumes
In General:
Vertical Cut:
 top
  bottom 
A  

dx


 function  function
a
b
ab
Horizontal Cut:
 right
  left

A  

dy


 function  function
c
d
cd
Find the area of the region bounded by
x  y2  4
and
Bounds? In terms of y: [-2,1]
Points: (0,-2), (3,1)
Right Function? x  4  y2
Left Function?
x2y
Area?
1
2
[(4

y
)  (2  y)]dy

2
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 Space
Volume of a blob:
Cross sectional area at
height h: A(h)
Volume =
Example
Solid 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
2
2
circle x  y  1 and every cross section
perpendicular to the x-axis is a square.
Bounds?
[-1,1]
Top Function?
y  1  x2
2
y


1

x
Bottom Function?
Length?

1  x2   1  x2
 2 1 x2

Find the volume of the solid whose bottom face is the
2
2
circle x  y  1 and every cross section
perpendicular to the x-axis is a square.
We use this length to find
the area of the square.
Length?  2 1 x2
Area?
2 1 x 
2
2




4 1 x 2
Volume?
1
2
4
1

x
dx

1
Find the volume of the solid whose bottom face is the
2
2
circle x  y  1 and every cross section
perpendicular to the x-axis is a square.
What does this shape look like?
1
Volume?


2
4
1

x
dx

1
Find the volume of the solid whose bottom face is the
circle x 2  y2  1 and every cross section
perpendicular to the x-axis is a circle with diameter
in the plane.
 2 1 x2
Length?
Area?
2 1  x2
2
Radius:

 1 x 
2
2
 1 x 2 
1
Volume?
2

1

x

dx

1
Using the half circle x  y  1 [0,1]
as the base slices perpendicular
to the x-axis are isosceles right
triangles.
2
Bounds?
[0,1]
Length?
 2 1 x2
Area?


1
2 1  x2 2 1  x2
2
1
Volume?
0
Visual?


2
2
1
x
dx


2
The base of the solid is the region between the curve
y  2 sin x 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.
Bounds?
[0,π]
Top Function?
y  2 sin x
Bottom Function?
Length?
Area of an equilateral triangle?
y0
2 sin x
Area of an Equilateral Triangle?
S
S
Sqrt(3)*S/2
S/2
S S/2
Area = (1/2)b*h
 3 
 1
3 2
   S 
S 
S
 2
 2 
4
The base of the solid is the region between the curve
y  2 sin x 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.
Bounds?
[0,π]
Top Function?
y  2 sin x
Bottom Function?
Length?
Area of an equilateral triangle?

Volume?

0
 3 2
 4 (S) 


3
(2 sin x )2 dx
4
2 sin x
3
(2 sin x )2
4
y0
Find the volume of the solid whose bottom face is the
circle x 2  y2  1 and every cross section
perpendicular to the x-axis is a square with diagonal in
the plane.
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?
S
S
D
S 2  S 2  D2
2S 2  D 2
2
D
S2 
2
D
S
2
Find the volume of the solid whose bottom face is the
circle x 2  y2  1 and every cross section
perpendicular to the x-axis is a square with diagonal in
the plane.
Length of Diagonal?  2 1 x2
Length of Side?
D
(S 
)
2
2 1  x2
 2 1 x2
2
Area?
2(1  x 2 )
1
Volume?


2
2
1

x
dx

1