x axis and the parabola is revolved about the line x 1 to generate the shape of a cake What is the volume of the cake DO NOW The Shell Method Section 73d The region enclosed by the ID: 500777
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Slide1
The region enclosed by the x-axis and the parabola
is revolved about the line
x
= –1 to
generate the shape of a cake. What is the volume of the cake?
DO NOWSlide2
The Shell Method
Section 7.3dSlide3
The region enclosed by the
x
-axis and the parabola
is revolved about the line
x
= –1 to
generate the shape of a cake. What is the volume of the cake?
x
= –1
Let’s first try to integrate with
respect to
y
Washers!!!
To find the inner and outer radii of our washer, we would
need
to get the equation of the original parabola in terms of
x
...
n
ot an easy task…Slide4
The region enclosed by the x-axis and the parabola
is revolved about the line
x
= –1 to
generate the shape of a cake. What is the volume of the cake?
Instead of slicing horizontally,
we will cut a series of
cylindrical
s
lices by cutting straight down
all the way around the solid.
The radii of the cylinders gradually increase, and the heights of
t
he cylinders follow the contour of the parabola: smaller to larger,
t
hen back to smaller.
When we unroll one of these
shells
, we have a rectangular
p
rism, with height equal to the height of the shell, length
e
qual to the circumference of the shell, and thickness equal
t
o .Slide5
The region enclosed by the x-axis and the parabola
is revolved about the line
x
= –1 to
generate the shape of a cake. What is the volume of the cake?
Volume of each shell:
Here, we have
To find the total volume of the solid, integrate with respect to
x
:Slide6
The region bounded by the curve , the x-axis, and the
line x = 4 is revolved about the
x-axis to generate a solid. Findt
he volume of the solid.
Let’s solve this problem first by integrating with respect to
x
:
Cross section area:
Volume:Slide7
The region bounded by the curve , the x-axis, and the
line x = 4 is revolved about the
x-axis to generate a solid. Findt
he volume of the solid.
Now let’s integrate with respect to
y
using our new technique:
Radius of each shell:
Height of each shell:
Limits of integration:
0 to 2
Thickness of each shell:Slide8
The region bounded by the curve , the x-axis, and the
line x = 4 is revolved about the
x-axis to generate a solid. Findt
he volume of the solid.
Now let’s integrate with respect to
y
using our new technique:
Volume:Slide9
Let’s redo a problem from a previous class, this time using shells…
Find the volume of the solid generated by revolving the region
bounded above by the square root function and below by the
identity function about the
y
-axis
Cross section area:
Volume:
Our previous solutionSlide10
Let’s redo a problem from a previous class, this time using shells…
Find the volume of the solid generated by revolving the region
bounded above by the square root function and below by theidentity function about the
y-axisShell radius:
Volume:
Shell height:
Shell thickness:Slide11
Let’s redo a problem from a previous class, this time using shells…
Find the volume of the solid generated by revolving the triangularregion bounded by the lines y
= 2x,
y = 0, and x = 1 about thel
ine x = 2.
R
r
Cross section area:
Volume:
Our previous solutionSlide12
Let’s redo a problem from a previous class, this time using shells…
Find the volume of the solid generated by revolving the triangularregion bounded by the lines y
= 2x,
y = 0, and x = 1 about thel
ine x = 2.
Shell radius:
Shell height:
Shell thickness:
Volume: