MATH1014 Calculus II

Tutorial 2 (1) Method of Slicing

(a) Cross-sectional Area A(x) perpendicular to the x-axis, The Volume 𝑽 = ∫ 𝑨(𝒙)𝒅𝒙𝒃

𝒂

(b) Cross-sectional Area A(y) perpendicular to the y-axis, The Volume 𝑽 = ∫ 𝑨(𝒚)𝒅𝒚𝒅

𝒄

(2) Disk / Washer Method

(a) Revolved about the x-axis,

The Volume = ∫ 𝝅𝒚𝟐𝒅𝒙𝒃

𝒂= ∫ 𝝅(𝒇(𝒙))𝟐𝒅𝒙

𝒃

𝒂 .

(b) Revolved about the y-axis,

The Volume = ∫ 𝝅𝒙𝟐𝒅𝒚𝒅

𝒄= ∫ 𝝅(𝒈(𝒚))𝟐𝒅𝒚

𝒅

𝒄 .

(3) Cylindrical Shell Method

(a) Revolved about the y-axis,

The Volume = ∫ 𝟐𝝅 𝒙⏟𝒓𝒂𝒅𝒊𝒖𝒔

(𝒇(𝒙) − 𝒈(𝒙))⏟ 𝒉𝒆𝒊𝒈𝒉𝒕

𝒅𝒙𝒃

𝒂 .

(b) Revolved about the x-axis,

The Volume = ∫ 𝟐𝝅 𝒚⏟𝒓𝒂𝒅𝒊𝒖𝒔

[𝒎(𝒚) − 𝒏(𝒚)⏟ 𝒉𝒆𝒊𝒈𝒉𝒕

]𝒅𝒚𝒃

𝒂 .

Example

Solution:

(4) Volume of Revolution about an axis other than the x-axis and y-axis

(a) Revolving about the line 𝒚 = 𝒉 .

Let 𝒚 = 𝒇(𝒙) be a function continuous on [a, b] and let S be the region bounded by

the curve 𝒚 = 𝒇(𝒙) and the line 𝒙 = 𝒂 , 𝒙 = 𝒃 and 𝒚 = 𝒉 . Then the volume of the

solid generated by revolving the region S one complete revolution about the line

𝒚 = 𝒉 is given by : 𝑽 = ∫ 𝝅(𝒚 − 𝒉)𝟐𝒅𝒙 = ∫ 𝝅(𝒇(𝒙) − 𝒉)𝟐𝒅𝒙𝒃

𝒂

𝒃

𝒂 .

(b) Revolving about the line 𝒙 = 𝒌 .

Let 𝒙 = 𝝋(𝒚) be a function continuous on [c, d] and let S be the region bounded by

the curve 𝒙 = 𝝋(𝒚) and the line 𝒚 = 𝒄 , 𝒚 = 𝒅 and 𝒙 = 𝒌 . Then the volume of the

solid generated by revolving the region S one complete revolution about the line

𝒙 = 𝒌 is given by : 𝑽 = ∫ 𝝅(𝒙 − 𝒌)𝟐𝒅𝒚 = ∫ 𝝅(𝝋(𝒚) − 𝒌)𝟐𝒅𝒚𝒃

𝒂

𝒅

𝒄 .

Exercises

Washers vs. shells method

1) Let R be the region bounded by the following curves. Let S be the solid generated

when R is revolved about the given axis. If possible, find the volume of S by both the

disk / washer and shell methods. Check that your results agree and state which

method is easiest to apply. For question 1(a) and 1(b)

(a) 2)2( 3 xy , 250 yandx ; revolved about the y -axis.

(1(a) Ans. = 500π)

(b) 4xxy ,𝒚 = 𝟎 ; revolved about the y -axis.

(1(b) Ans. =π/3)

2) (METHOD OF SLICING) The solid

with a circular base of radius 5 whose cross

sections perpendicular to the base and

parallel to the x -axis are equilateral

triangles. (2. Ans=𝟓𝟎𝟎√𝟑

𝟑)

3) The solid whose base is the region bounded

by2xy , and the line 1y and whose

cross sections perpendicular to the base and parallel to the x -axis are squares.

(ANS.=2)

4) (Stewart p.438)

A wedge is cut out of a circular cylinder of radius 4

by two planes. One plane is perpendicular to the

axis of the cylinder. The other intersects the first at

an angle of 𝟑𝟎𝟎 along a diameter of the cylinder.

Find the volume of the wedge.

5) (Stewart Ex.6.2#55)

The base 𝑺 of is an elliptical region with boundary curve

𝟗𝒙𝟐 + 𝟒𝒚𝟐 = 𝟑𝟔. Cross-sections perpendicular to the x–axis are isosceles right

triangles with hypotenuse in the base.

6) (Stewart Ex.6.2#49) Find the volume of a cap of sphere with

radius 𝒓 and height 𝒉.

7) (Stewart Ex.6.2#12)

Find the volume of the solid obtained by rotating the region

bounded by the curves 𝒚 = 𝒆−𝒙 , 𝒚 = 𝟏 , 𝒙 = 𝟐 ; about the

line 𝒚 = 𝟐 . Sketch the region, the solid, and a typical disk or washer.

8) (Stewart Ex.6.2#33(b))

Set up an integral for the volume of the solid obtained by rotating the region bounded

by the curves 𝒙𝟐 + 𝟒𝒚𝟐 = 𝟒 about the line 𝒙 = 𝟐 . Then use your calculator to

evaluate the integral correct to five decimal places.

9) (Stewart Ex.6.3#12)

Use the method of cylindrical shells to find the volume of the solid obtained by

rotating the region bounded by the curves

𝒙 = 𝟒𝒚𝟐 − 𝒚𝟑 , 𝒙 = 𝟎 ; about the x-axis.

10) (Stewart Ex.6.3#17)

Use the method of cylindrical shells to find the volume generated by rotating the

region bounded by the curves 𝒚 = 𝟒𝒙 − 𝒙𝟐 , 𝒚 = 𝟑 ; about the axis 𝒙 = 𝟏 .

11) (Stewart Ex.6.3#19)

Use the method of cylindrical shells to find the volume generated by rotating the

region bounded by the curves 𝒚 = 𝒙𝟑 , 𝒚 = 𝟎 , 𝒙 = 𝟏 ; about the axis 𝒚 = 𝟏 .

12) (Stewart Ex.6.3#24)

(a) Set up an integral for the volume of the solid obtained by rotating the region

bounded by the curves 𝒙 = 𝒚 , 𝒚 =𝟐𝒙

𝟏+𝒙𝟑 ; about the axis 𝒙 = −𝟏 .

(b) Use your calculator to evaluate the integral correct to five decimal places.

13) (Stewart Ex.6.3#26)

(a) Set up an integral for the volume of the solid obtained by rotating the region

bounded by the curves 𝒙𝟐 − 𝒚𝟐 = 𝟕 , 𝒙 = 𝟒 ; about the axis 𝒚 = 𝟓 .

(b) Use your calculator to evaluate the integral correct to five decimal places.

14) (Stewart Ex.6.3#39)

The region bounded by the curves 𝒚𝟐−𝒙𝟐 = 𝟏 , 𝒚 = 𝟐 is rotated about the x-axis.

Find the volume of the resulting solid by any method.

Graph

15) (Stewart Ex.6.2#45)

(a) If the region shown in the figure is rotated

about the x-axis to form a solid, use the

Midpoint Rule with 𝒏 = 𝟒 to estimate the

volume of the solid.

(b) Estimate the volume if the region is rotated about the y-axis. Again use the

Midpoint Rule with 𝒏 = 𝟒 .

16) (Stewart Ex.6.3#28)

If the region shown in the figure is rotated about

the y-axis to form a solid, use the Midpoint Rule

with 𝒏 = 𝟓 to estimate the volume of the solid.