Transcript Calculus Notes Ch 6.2 •Volumes by slicing can be found by adding up each slice of the solid as the thickness of.

Calculus Notes Ch 6.2
•Volumes by slicing can be found by adding up each
slice of the solid as the thickness of the slices gets
smaller and smaller, in other words
the number of slices goes to infinity.
•To find the volume of each slice of the pyramid you
would find the area of each square then multiply the
area by the thickness of the slice. The thickness would
be dx because we are slicing with respect to the x-axis.
Next you would add the volumes of each slice together
to find the total volume.
•This would be a Riemann sum with the limit as n the
number of slices going to infinity.
• Remember the from Ch 5.5
b
n
A
lim  f ( x )x   f ( x)dx
*
k
max x 0 k 1
k
a
• Only now instead of f(x*) we will be summing the
areas of the base of each slice. So change
f(x*)in the Riemann Sum to A(x*)which
represents the area of the base.
V
n
lim
b
*
A
(
x
 k )xk   A( x)dx
max x 0 k 1
a
Here is an example using a
cylinder
Find the volume of the cylinder
using the formula and slicing
with respect to the x-axis.
A = r2 A = 22 = 4
6
 4 dx
2
 4x

6
2
 4 (6)  4 (2)  24  8  16
• Now use the formula you learned in
Geometry to find the area
• V= r2 h=(22)(4) = 16 
You can also work problems like
these with respect to the y-axis.
• Volume of solids of revolution using
disks
• If you take the area under the line y = x
fromIf0you
to rotate
4 it will
like the
thislook
area around
thediagram
x-axis
below
it will form a cone (see below diagram)
4
4
• Now use the formula below to find the
volume of the 3-D figure formed by
rotating around the x-axis.
• http://www.plu.edu/~heathdj/java/calc2/Solid.html
• This method is called disks when revolved
around the x-axis ( note: it is sliced with
respect to the x-axis and is revolved
around the x-axis)
• Formula for finding the volume of the solid
formed when f(x) is revolved around the xaxis
• Each slice is a circle, the formula for the
area of a circle is A= r2. The radius is the yvalue of the function AKA f(x). So, the area
formula becomes A= (f(x))2
• Substitute into the formula
b
b
 A( x) dx     f ( x) dx
2
a
a
4
• Equation y = x from 0 to 4
With geometry V = 1/3  (42)(4)=64/3
• With Calculus
4
  (x
0
2
) dx 
4
x 
 64  64
      0 
3 0
3
 3

3
Your turn
p. 410 # 6. find the volume of the solid that
results when the region enclosed by the
given curves is revolved around the x-axis


y = sec x, x = 4 , x = 3 , y = 0
b
2



f
(
x
)
dx

Use the formula

a

3
  sec x dx   tan x
2
3
4
4




   tan  tan 
3
4


 3 1   3  