math 1014 tutorial notes 2 - masxliang.people.ust.hkmasxliang.people.ust.hk/math1014 tutorial notes...

MATH1014 Calculus II (2019 Spring) Tutorial Notes 2(Phyllis LIANG)

of 5

MATH 1014 Tutorial Notes 2

Topics covered in tutorial 02:

1. Volume by slicing

2. Volume of revolution

1. Volume by slicing

What you need to know:

• Calculate the volume by slicing method

Volume by slicing Let 𝑆 be a solid that lies between 𝑥 = 𝑎 and 𝑥 = 𝑏. If the cross-section area of 𝑆 in the plane 𝑃𝑋,

through 𝑥 and perpendicular to the 𝑥 − 𝑎𝑥𝑖𝑠, is 𝐴(𝑥), where 𝐴 is a continuous function, then the volume of 𝑆 is

𝑽 = 𝐥𝐢𝐦𝒏→∞ ∑ 𝑨(𝒙𝒊∗)𝒏

𝒊=𝟏 ∆𝒙 = ∫ 𝑨(𝒙)𝒅𝒙𝒃

𝒂.

Remark:

(a) Cross-sectional area 𝐴(𝑥) perpendicular to the 𝒙 − 𝒂𝒙𝒊𝒔, then the volume 𝑽 = ∫ 𝑨(𝒙)𝒅𝒙𝒃

𝒂

(b) Cross-sectional area 𝐴(𝑦) perpendicular to the 𝒚 − 𝒂𝒙𝒊𝒔, then the volume 𝑽 = ∫ 𝑨(𝒚)𝒅𝒚𝒅

𝒄

Example 2.1 The base of a certain solid is the area bounded above by the graph of 𝑦 = 9, and below by the graph of 𝑦 = 36𝑥2, cross-sections perpendicular to the x-axis are squares. Find the volume of the solid.

Example 2.2 The base of a certain solid is an equilateral triangles with altitude 6, cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid.


of 5

2. Volume of revolution

What you need to know:

• Calculate the volume of revolution by Washer/Disk Method

• Calculate the volume of revolution by Cylindrical Shell/Shell Method

• Calculate the volume of revolution about an axis other than 𝑥 − 𝑎𝑥𝑖𝑠, and 𝑦 − 𝑎𝑥𝑖𝑠.

Washer /Disk Method:

Revolved about the 𝒙 − 𝒂𝒙𝒊𝒔:

𝑽 = ∫ 𝝅𝒚𝟐𝒅𝒙 =𝒃

𝒂

∫ 𝝅(𝒇(𝒙))𝟐

𝒅𝒙𝒃

𝒂

Revolved about the 𝒚 − 𝒂𝒙𝒊𝒔:

𝑽 = ∫ 𝝅𝒙𝟐𝒅𝒚 =𝒅

𝒄

∫ 𝝅(𝒈(𝒚))𝟐

𝒅𝒚𝒅

𝒄

Example 2.3 Find the volume of the solid obtained by rotating about the 𝑥 − 𝑎𝑥𝑖𝑠 the region enclosed by 𝑦 = 𝑒𝑥 + 4, 𝑦 = 0, 𝑥 = 0,and 𝑥 = 0.3.

Example 2.4 Find the volume of the solid obtained by rotating the region bounded by 𝑦 = 𝑥3, 𝑦 = 8, 𝑎𝑛𝑑 𝑥 = 0 about the 𝑦 − 𝑎𝑥𝑖𝑠.


of 5

Cylindrical Shell Method:

Revolved about the 𝒙 − 𝒂𝒙𝒊𝒔:

𝑽 = ∫ 𝟐𝝅𝒚𝒈(𝒚)𝒅𝒚𝒅

𝒄

Revolved about the 𝒚 − 𝒂𝒙𝒊𝒔:

𝑽 = ∫ 𝟐𝝅𝒙𝒇(𝒙)𝒅𝒙𝒃

𝒂

Example 2.5 Find the volume of the solid obtained by rotating about the 𝑦 − 𝑎𝑥𝑖𝑠 the region bounded by 𝑦 = 0,

and 𝑦 = 2𝑥2 − 𝑥3.


of 5

Example 2.6 Find the volume of the solid obtained by rotating about the 𝑦 − 𝑎𝑥𝑖𝑠 the region between 𝑦 = 𝑥, 𝑦 = 𝑥2.

The volume of revolution about an axis other than the 𝐱 − 𝐚𝐱𝐢𝐬, and 𝐲 − 𝐚𝐱𝐢𝐬:

Example 2.7 Find the volume of the solid obtained by rotating the region bounded by 𝑦 = 0, 𝑦 = √𝑥 − 1, 𝑥 = 5 about the line 𝑦 = 7.

Example 2.8 Set up an integral that represents the volume of the solid obtained by rotating the region

bounded by 𝑥 = 0, 𝑦 = 0, 𝑥 = √cos (𝑦), 0 ≤ 𝑦 ≤ 𝜋 about 𝑦 = 3.


of 5

Summary: Disk/Washer and Shell Methods:

math 1014 tutorial notes 2 - masxliang.people.ust.hkmasxliang.people.ust.hk/math1014 tutorial notes...

Documents