Volumes of Known Cross-Sections - SLC Home Page

Math 201-203-RE
Calculus II Exercises
Volumes of Known Cross-Sections
Filename: E15) @ Volumes Of Known Cross-Sections.Doc
Carl F. Gauss
a) Determine the volume of a pyramid 12 feet high with a rectangular base whose width is 3 feet and length is 6 feet
and for which parallel cross section to the base is a rectangle whose length is twice as large as its width.
b)
When viewed from above, a swimming pool has the shape of the ellipse
2 2
y
900 400
x
+ = 1 . The cross sections of the pool
perpendicular to the ground and parallel to the y-axis are squares. If the units are in feet, what is the volume of the
pool?
c) A log having the shape of a right circular cylinder of radius 3 is lying on its
side (see Figure E15-01). A wedge is removed from the log by making a
vertical cut and another cut at an angle of 45°, both cuts intersect at the
center of the log. Find the volume of the wedge.
Figure E15-01
d)
e)
f)
g)
The tank of a milk delivery truck has the shape of a cylinder 12 meters long with a 2 meters diameter. Determine the
number of liter of milk contained if the tank is filled up to 1.5 meters (1000 liters = 1 m 3 ).
2 2
Find the volume of the solid whose base is bounded by the circle x + y = 9 with the indicated cross sections taken
perpendicular to the x-axis.
a) Squares b) Isosceles right triangles (hypotenuse on the base)
Find the volume of the solid whose base is bounded by the graphs of y = x + 1 and y = x − 1 with the indicated
cross sections taken perpendicular to the x-axis.
a) Squares b) Rectangles of height 1
The base of a solid is the hyperbola
2 2
y
4 9
x − = 1 for 2 ≤ x ≤ 5 , and the
cross sections perpendicular to the x-axis are squares.
2
Figure E18-02
h) The base of a solid is a isosceles right triangle whose legs are each 4 units
long. Each cross section perpendicular to a side is a semicircle.
Figure E18-03
i)
j)
2 2
Find the volume of the solid whose base is bounded by the circle x + y = 4 , with the indicated cross sections taken
perpendicular to the x-axis.
a) Squares b) Equilateral triangles
c) Semicircles d) Isosceles right triangles (hypotenuse on the base)
Find the volume of the solid whose base is bounded by y = x 3 , y = 0, and x = 1. Find the volume of the solid for the
following cross sections (taken perpendicular to the y-axis).
a) Squares b) Semicircles c) Equilateral triangles

a)
ANSWERS:
3
72 ft b)
3
64000 ft
c)
3
18 u d) ≈ 30329 liters
e) a)
3
144 u b)
3
36 u
f) a)
81 3
10 u b) 9 3
2 u
g) 3
243 u h)
8π
3
3 u
i) a)
128
3
3
u b)
32 3
3
3
u c)
16
3
3
π
u
d)
32 3
3 u
j) a)
1 3
10 u b) 3
80 u
π
c)
3 3
40 u