7-2: Volume Objectives: Assignment:

7-2: Volume
Objectives:
Assignment:
1. To find the volume of a
solid of revolution using
disks and washers
• P. 463-465: 1-11 odd, 12,
23, 27, 33, 45, 49, 53, 55
2. To find the volume of a
solid with known cross
sections
• P. 465-466: 61-63
• Homework Supplement
Warm Up 1
Find the volume of
the given figure in
cubic units.
Cavalieri’s Principle
If two solids have the same height and the
same cross-sectional area at every level,
then they have the same volume.
All 3 of these shapes have the same volume.
𝑉 = 𝐵ℎ
Warm Up 2
Draw and name the 3-D solid of revolution
formed by revolving the given 2-D shape
around the y-axis.
Solids of Revolution
Draw and name the 3-D solid of revolution
formed by revolving the given 2-D shape
around the y-axis.
Sphere
Hemisphere
Torus
Warm Up 3
Draw and name the
3-D solid of
revolution formed
by revolving the
given 2-D shape
around the
indicated axis.
What is its volume?
𝑟
ℎ
Solids of Revolution
Draw and name the
3-D solid of
revolution formed
by revolving the
given 2-D shape
around the
indicated axis.
What is its volume?
Cylinder
𝑉 = 𝜋𝑟 2 ℎ
Objective 1
You will be able to find the
volume of a solid of revolution
using disks and washers
Disk Method
Much like finding the
area between two
curves, to find the
volume of a solid of
revolution, we first
consider a representative
rectangle.
Disk Method
Revolving this
rectangle around an
axis generates a
disk.
Volume of Disk:
𝑉 = 𝜋𝑟 2 ℎ
𝑉 = 𝜋𝑅2 ∆𝑥
𝑅 is a function of 𝑥
Disk Method
We can approximate
the volume of the
solid by summing 𝑛
such disks in a
Riemann sum.
𝑛
𝑉≈
𝜋 𝑅 𝑥𝑖
𝑖=1
2 ∆𝑥
Disk Method
Taking the limit as
𝑛 → ∞ of this sum
yields the exact
volume of the solid.
𝑛
𝑉 = lim
𝑛→∞
𝜋 𝑅 𝑥𝑖
𝑖=1
𝑏
𝑉=
𝜋𝑅 𝑥
𝑎
2
𝑑𝑥
2 ∆𝑥
Disk Method
Taking the limit as
𝑛 → ∞ of this sum
yields the exact
volume of the solid.
𝑛
𝑉 = lim
𝑛→∞
𝜋 𝑅 𝑥𝑖
𝑖=1
𝑏
𝑉=𝜋
𝑅 𝑥
𝑎
2
𝑑𝑥
2 ∆𝑥
Vertical vs. Horizontal Disks
Depending on the axis of rotation, you may
have vertical or horizontal disks.
Exercise 1
Find the volume of the
solid formed by
revolving the region
bounded by 𝑦 = sin 𝑥
and the 𝑥-axis on the
interval 0, 𝜋 about the
𝑥-axis.
Exercise 1
Find the volume of the
solid formed by
revolving the region
bounded by 𝑦 = sin 𝑥
and the 𝑥-axis on the
interval 0, 𝜋 about the
𝑥-axis.
Exercise 2
Find the volume of the solid formed by
revolving the region bounded by 𝑦 = 𝑥 2 and
the 𝑥-axis on the interval 0,2 about the
𝑥-axis.
Exercise 3
Find the volume of the solid formed by
1
revolving the region bounded by 𝑦 = 𝑥 − 1,
2
𝑦 = 0, 𝑦 = 2, and 𝑥 = 0 about the 𝑦-axis.
Exercise 4
Find the volume of the
solid formed by revolving
the region bounded by
𝑓 𝑥 = 2 − 𝑥 2 and
𝑔 𝑥 = 1 about the line
𝑦 = 1.
Exercise 4
Find the volume of the
solid formed by revolving
the region bounded by
𝑓 𝑥 = 2 − 𝑥 2 and
𝑔 𝑥 = 1 about the line
𝑦 = 1.
Exercise 5
Draw and name the
3-D solid of
revolution formed
by revolving the
given 2-D shape
about the indicated
axis. What is its
volume?
Exercise 5
Draw and name the
3-D solid of
revolution formed
by revolving the
given 2-D shape
about the indicated
axis. What is its
volume?
Volume of Washer:
𝑉 = 𝜋𝑅2 𝑤 − 𝜋𝑟 2 𝑤
𝑉 = 𝜋 𝑅2 − 𝑟 2 𝑤
Outer Radius
Inner radius
Washer Method
Sometimes
revolving a 2-D
shape around an
axis produces a
solid with a hole.
As before, we still
need to consider a
representative
rectangle.
Washer Method
However, rotating
this rectangle about
an axis produces a
washer.
𝑉 = 𝜋 𝑅2 − 𝑟 2 ∆𝑥
Washer Method
The volume can be
approximated by
summing 𝑛 such
washers.
𝑛
𝑉≈
𝜋 𝑅 𝑥𝑖
𝑖=1
2
− 𝑟 𝑥𝑖
2
∆𝑥
Washer Method
Taking the limit as
𝑛 → ∞ yields the
volume of the solid.
𝑛
𝑉 = lim
𝑛→∞
𝜋 𝑅 𝑥𝑖
𝑖=1
2
− 𝑟 𝑥𝑖
2
∆𝑥
Washer Method
Taking the limit as
𝑛 → ∞ yields the
volume of the solid.
Outer Radius
Inner radius
𝑏
𝑉=
𝜋 𝑅 𝑥
𝑎
2
− 𝑟 𝑥
2
𝑑𝑥
Exercise 6
Find the volume of
the solid formed by
revolving the region
bounded by the
graphs of 𝑦 = 𝑥
and 𝑦 = 𝑥 2 about
the 𝑥-axis.
Exercise 6
Find the volume of
the solid formed by
revolving the region
bounded by the
graphs of 𝑦 = 𝑥
and 𝑦 = 𝑥 2 about
the 𝑥-axis.
Exercise 7
Find the volume of
the solid formed by
revolving the region
bounded by the
graphs of 𝑦 = 𝑥 2 + 1,
𝑦 = 0, 𝑥 = 0, and
𝑥 = 1 about the 𝑦axis.
Exercise 7
Find the volume of
the solid formed by
revolving the region
bounded by the
graphs of 𝑦 = 𝑥 2 + 1,
𝑦 = 0, 𝑥 = 0, and
𝑥 = 1 about the 𝑦axis.
Objective 2
You will be able to find the volume
of a solid with known cross
sections
Other Cross Sections
Finding the volume of a solid of revolution
can be extended to include solids of any
similar cross sectional shape, assuming you
can find its area.
Common Cross
Sections:
Squares
Rectangles
Triangles
Trapezoids
Semicircles
Other Cross Sections
To find the volume of these solids, find 𝐴 𝑥 ,
the area of one cross section. Multiplying
this by its width ∆𝑥 gives the volume of one
cross section.
Other Cross Sections
The volume can
then be
approximated by
summing 𝑛 of
these cross
sections in a
Riemann sum.
𝑛
𝑉≈
𝐴 𝑥𝑖 ∆𝑥
𝑖=1
Other Cross Sections
Finally, taking the
limit as 𝑛 → ∞
yields the exact
volume of the
solid.
𝑛
𝑉 = lim
𝑛→∞
𝐴 𝑥𝑖 ∆𝑥
𝑖=1
𝑏
𝑉=
𝐴 𝑥 𝑑𝑥
𝑎
Exercise 8
Consider a region 𝑅
bounded below by the
curve 𝑓 𝑥 = 𝑥 2 − 1
and above by the
curve 𝑔 𝑥 = 𝑥 + 1.
Find the area of 𝑅.
Consider a region 𝑅
bounded below by the
curve 𝑓 𝑥 = 𝑥 2 − 1 and
above by the curve 𝑔 𝑥 =
𝑥 + 1. Region 𝑅 is the
base of a solid. Find the
volume of the solid formed
by square cross sections
perpendicular to the 𝑥axis.
𝑔 𝑥 −𝑓 𝑥
Exercise 9
𝑔 𝑥 −𝑓 𝑥
Exercise 10
Consider a region 𝑅
bounded below by the
curve 𝑓 𝑥 = 𝑥 2 − 1 and
above by the curve 𝑔 𝑥 =
𝑥 + 1. Region 𝑅 is the base
of a solid. Find the volume
of the solid formed by
semicircular cross sections
perpendicular to the 𝑥-axis.
1
𝑔 𝑥 −𝑓 𝑥
2
Consider a region 𝑅
bounded below by the
curve 𝑓 𝑥 = 𝑥 2 − 1 and
above by the curve 𝑔 𝑥 =
𝑥 + 1. Region 𝑅 is the
base of a solid. Find the
volume of the solid formed
by equilateral triangular
cross sections
perpendicular to the 𝑥-axis.
3
𝑔 𝑥 −𝑓 𝑥
2
Exercise 11
𝑔 𝑥 −𝑓 𝑥
Consider a region 𝑆
bounded by 𝑦 = 𝑒 𝑥
and 𝑦 = 𝑥 + 2.
Region 𝑆 is the base
of a solid. Find the
volume of the solid
formed by squares
cross sections
perpendicular to the
𝑦-axis.
ln 𝑦 − 𝑦 − 2
Exercise 12
ln 𝑦 − 𝑦 − 2
Exercise 13: AP FRQ
Let 𝑓 and 𝑔 be functions
defined by
2 −2𝑥
𝑥
𝑓 𝑥 =1+𝑥+𝑒
and
𝑔 𝑥 = 𝑥 4 − 6.5𝑥 2 + 6𝑥 + 2.
Let 𝑅 and 𝑆 be the two
regions enclosed by the
graphs of 𝑓 and 𝑔 shown in
the figure.
Region 𝑆 is the base
of a solid whose
cross sections
perpendicular to the
𝑥-axis are squares.
Find the volume of
the solid.
Exercise 14: AP FRQ
Let 𝑓 and 𝑔 be functions
defined by
2 −2𝑥
𝑥
𝑓 𝑥 =1+𝑥+𝑒
and
𝑔 𝑥 = 𝑥 4 − 6.5𝑥 2 + 6𝑥 + 2.
Let 𝑅 and 𝑆 be the two
regions enclosed by the
graphs of 𝑓 and 𝑔 shown in
the figure.
Let ℎ be the vertical
distance between
the graphs of 𝑓 and
𝑔 in the region 𝑆.
Find the rate at
which ℎ changes
with respect to 𝑥
when 𝑥 = 1.8.
Exercise 15: AP FRQ
Let 𝑓 and 𝑔 be functions
defined by
2 −2𝑥
𝑥
𝑓 𝑥 =1+𝑥+𝑒
and
𝑔 𝑥 = 𝑥 4 − 6.5𝑥 2 + 6𝑥 + 2.
Let 𝑅 and 𝑆 be the two
regions enclosed by the
graphs of 𝑓 and 𝑔 shown in
the figure.
Find the volume of
the solid formed by
rotating region 𝑅
about the 𝑥-axis.
7-2: Volume
Objectives:
Assignment:
1. To find the volume of a
solid of revolution using
disks and washers
• P. 463-465: 1-11 odd, 12,
23, 27, 33, 45, 49, 53, 55
2. To find the volume of a
solid with known cross
sections
• P. 465-466: 61-63
• Homework Supplement