Volume By Slicing
AP Calculus
Volume
Volume = the
sum of the
quantities in each
layer
x
x
x
x
y
x-axis
Volume by Cross Sections
x
n

thickness(h)
BE7250
Axial (cross sectional) magnetic resonance image of a brain with a
large region of acute infarction, formation of dying or dead tissue,
with bleeding. This infarct involves the middle and posterior
cerebral artery territories.
Credit: Neil Borden / Photo Researchers, Inc.
Volume by Slicing
(Finding the volume of a solid built on the base in the x – y plane)
METHOD:
1.) Graph the “BASE”
2.) Sketch the line segment across the base.
That is the representative slice “n”
Use “n” to find: a.) x or  y (Perpendicular to axis)
b.) the length of “n”
3.) Sketch the “Cross Sectional Region” - the shape of the slice (in 3-D )
from Geometry V = B*h
h = or is the thickness of the slice
B = the Area of the cross section
4.) Find the area of the region
5) Write a Riemann’s Sum for the total Volume of all the Regions
Example 1: The base of a solid is the region in the x-y plane bounded by the graph
x  y2  3
and the y – axis. Find the volume of the solid if every cross section
by a plane perpendicular to the x-axis is a square.
Base
y

x


Cross -Section

Example 2: The base of a solid is the region in the x-y plane bounded by the graph
x  y 2  3 and the y – axis. Find the volume of the solid if every cross section
by a plane perpendicular to the x-axis is an Isosceles Rt. Triangle (leg on the base).
y

x



Some Important Area Formulas
Square-
Square-
side on base
diagonal on base
Isosceles
rt Δ
Isosceles
rt Δ
leg on base
hypotenuse on base
Equilateral Δ
EXAMPLE #4/406
The solid lies between planes perpendicular to the x - axis at x = -1 and
x = 1 . The cross sections perpendicular to the x – axis are circular disks
whose diameters run from the parabola y = x2
to the parabola y = 2 – x2.
Volumes of Revolution:
Disk and Washer Method
AP Calculus
Volume of Revolution: Method
Lengths of Segments:
In revolving solids about a line, the lengths of several segments are needed for the radii of
disks, washers, and for the heights of cylinders.
A). DISKS AND WASHERS
1) Shade the region in the first quadrant (to be rotated)
2) Indicate the line the region is to be revolved about.
3) Sketch the solid when the region is rotated about the indicated line.
4) Draw the representative radii, its disk or washer and give their lengths.
<<REM: Length must be positive! Top – Bottom or Right – Left >>
Ro =
outer radius
ri = inner radius
Disk Method
Rotate the region bounded by
f(x) = 4 – x2 in the first quadrant about the y - axis
The region is _______________ _______ the axis of rotation.
The Formula:
The formula is based on the
_____________________________________________
Washer Method
Rotate the region bounded by
f(x) = x2, x = 2 ,
and
y=0
about the
y - axis
The region is _______________ __________ the axis of rotation.
The Formula:
The formula is based on
_____________________________________________
Disk Method
Rotate the region bounded by
f(x) = 2x – 2 , x = 4 ,
and
y=0
about the line
x=4
The region is _______________ _______ the axis of rotation.
Washer Method
Rotate the region bounded by
f(x) = -2x + 10 , x = 2 ,
and
y=0
about the
y - axis
The region is _______________ __________ the axis of rotation.
Example 1:
The region is bounded by
the x-axis, and the y-axis
y  4  x2
Rotated about:
a) The x-axis
b) The y-axis
c) x = 3
d) y = 4
Example 2:
The region is bounded by:
f(x) = x and g(x) = x2
in the first quadrant
Rotated about:
a)
b)
c)
d)
the x-axis
the y-axis
x=2
y=2
Example 2: The base of a solid is the region in the x-y plane bounded by the graph
x  y 2  3 and the x – axis. Find the volume of the solid if every cross section by
a plane perpendicular to the x-axis is an Isosceles Rt. Triangle (leg on the base).