MATH1014 Calculus II
Tutorial 2 (1) Method of Slicing
(a) Cross-sectional Area A(x) perpendicular to the x-axis, The Volume π½ = β« π¨(π)π ππ
π
(b) Cross-sectional Area A(y) perpendicular to the y-axis, The Volume π½ = β« π¨(π)π ππ
π
(2) Disk / Washer Method
(a) Revolved about the x-axis,
The Volume = β« π πππ ππ
π= β« π (π(π))ππ π
π
π .
(b) Revolved about the y-axis,
The Volume = β« π πππ ππ
π= β« π (π(π))ππ π
π
π .
(3) Cylindrical Shell Method
(a) Revolved about the y-axis,
The Volume = β« ππ πβπππ πππ
(π(π) β π(π))β ππππππ
π ππ
π .
(b) Revolved about the x-axis,
The Volume = β« ππ πβπππ πππ
[π(π) β π(π)β ππππππ
]π ππ
π .
Example
Solution:
(4) Volume of Revolution about an axis other than the x-axis and y-axis
(a) Revolving about the line π = π .
Let π = π(π) be a function continuous on [a, b] and let S be the region bounded by
the curve π = π(π) and the line π = π , π = π and π = π . Then the volume of the
solid generated by revolving the region S one complete revolution about the line
π = π is given by : π½ = β« π (π β π)ππ π = β« π (π(π) β π)ππ ππ
π
π
π .
(b) Revolving about the line π = π .
Let π = π(π) be a function continuous on [c, d] and let S be the region bounded by
the curve π = π(π) and the line π = π , π = π and π = π . Then the volume of the
solid generated by revolving the region S one complete revolution about the line
π = π is given by : π½ = β« π (π β π)ππ π = β« π (π(π) β π)ππ ππ
π
π
π .
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. = 500Ο)
(b) 4xxy ,π = π ; revolved about the y -axis.
(1(b) Ans. =Ο/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=πππβπ
π)
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 πππ along a diameter of the cylinder.
Find the volume of the wedge.
5) (Stewart Ex.6.2#55)
The base πΊ of is an elliptical region with boundary curve
πππ + πππ = ππ. Cross-sections perpendicular to the xβaxis 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 π and height π.
7) (Stewart Ex.6.2#12)
Find the volume of the solid obtained by rotating the region
bounded by the curves π = πβπ , π = π , π = π ; about the
line π = π . 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 ππ + πππ = π about the line π = π . 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
π = πππ β ππ , π = π ; 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 π = ππ β ππ , π = π ; about the axis π = π .
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 π = ππ , π = π , π = π ; about the axis π = π .
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 π = π , π =ππ
π+ππ ; about the axis π = βπ .
(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 ππ β ππ = π , π = π ; about the axis π = π .
(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 ππβππ = π , π = π 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 π = π 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 π = π .
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 π = π to estimate the volume of the solid.
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