Transcript Document
Finding Volumes
In General:
Vertical Cut:
top
bottom
A
dx
function function
a
b
ab
Horizontal Cut:
right
left
A
dy
function function
c
d
cd
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?
x2y
Area?
1
2
[(4
y
) (2 y)]dy
2
xy2
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?
y0
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
y0
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