Announcements

Calculus IIWednesday, February 8th

I WebAssign 4 due Monday Feb 13

I Midterm 1 is Wednesday Feb 15

I Problem Set 4 due Wednesday Feb 22

Today: Sec. 7.2: Disk Method

Find the volume of a solid of revolution about any axisusing the disk/washer method.

Next Class: Sec. 7.3: Shell Method

Cherveny Feb 8 Math 1103: Calculus II

Intro to Disk Method

TheoremIf the region between f (x) and the x-axis on [a, b] is rotated aboutthe x-axis, the resulting solid of revolution has volume

V =

∫ b

aπ(f (x))2 dx

Proof.

Cherveny Feb 8 Math 1103: Calculus II

Example

Example

Sketch and calculate the volume of the solid formed by revolving

f (x) =1

x2

about the x-axis on [1, 4].

Answer:

V =

∫ 4

1π

(1

x2

)2

dx = − π

3x3

∣∣∣∣41

=21π

64

Cherveny Feb 8 Math 1103: Calculus II

Practice

1. Consider the region R between the curves

y =10

xx = 0

y = 2

y = 10

(a) Find the area of R by integrating in the x-direction.(b) Find the area of R by integrating in the y -direction.

2. Show the volume of a sphere of radius R is

V =4

3πR3

by using the disk method.

Cherveny Feb 8 Math 1103: Calculus II

Practice Answers

1. (a) ∫ 1

0

(10− 2) dx +

∫ 5

1

(10

x− 2

)dx = · · · = 10 ln 5

(b) ∫ 10

2

10

ydy = 10 ln 5

2. Let y =√R2 − x2, defining the top half of a circle radius R.

Revolve this about x-axis to get the sphere.

V =

∫ R

−Rπ(√

R2 − x2)2

dx = 2

∫ R

0π(R2 − x2

)dx

= 2π

(R2x − x3

3

) ∣∣∣∣R0

=4

3πR3

Cherveny Feb 8 Math 1103: Calculus II

Washer Method

TheoremIf the region between f (x) and g(x), with f (x) > g(x), is rotatedabout the x-axis, the resulting solid of revolution has volume

V = π

∫ b

a

[f (x)2 − g(x)2

]dx

Cherveny Feb 8 Math 1103: Calculus II

Example

Example

Find the volume of the solid formed by revolving the regionbounded by

y = 2x

y = x2 + 4

x = 0

x = 3

around the x-axis?

Answer:

V = π

∫ 3

0(x2 + 4)2 − (2x)2 dx =

663π

5

Cherveny Feb 8 Math 1103: Calculus II

General Disk/Washer Method

TheoremIn general, when a the region is rotated about an axis, the resultingsolid of revolution has volume

V = π

∫ b

a(outer radius)2 − (inner radius)2 dx

Cherveny Feb 8 Math 1103: Calculus II

Practice II

1. Consider the region R bounded by the curves y = ex , y = 1,and x = 1. Write an integral giving...

(a) the area of the region R.(b) the volume of the solid formed by revolving R around y = 0.(c) the volume of the solid formed by revolving R around the axis

y = 7.(d) the volume of the solid whose base is R and whose cross

sections perpendicular to the x-axis are squares.

2. Consider the region R bounded by the curves x = −y2 + 3and x = 2. Write an integral giving...

(a) the area of the region R.(b) the volume of the solid formed by revolving R around x = 0.(c) the volume of the solid formed by revolving R around the axis

x = −3.(d) the volume of the solid whose base is R and whose cross

sections perpendicular to the y -axis are equilateral triangles(!)

Cherveny Feb 8 Math 1103: Calculus II

Practice II Answers

1. (a) A =

∫ 1

0

ex − 1 dx

(b) V = π

∫ 1

0

(ex)2 − (1)2 dx

(c) V = π

∫ 1

0

(7− 1)2 − (7− ex)2 dx

(d) V =

∫ 1

0

(ex − 1)2 dx

2. (a) A =

∫ 1

−1−y2 + 3− 2 dy

(b) V = π

∫ 1

−1(−y2 + 3)2 − (2)2 dy

(c) V = π

∫ 1

−1(−y2 + 3 + 3)2 − (2 + 3)2 dy

(d) Area Equil. ∆ =√34 (side)2, so V =

∫ 1

−1

√3

4(−y2 + 3− 2)2 dy

Cherveny Feb 8 Math 1103: Calculus II