Day 5: Arc Length and Surface Area

Report
ARC LENGTH AND SURFACE AREA
Compiled by Mrs. King
START WITH SOMETHING EASY
The length of the line segment joining points (x0,y0)
and (x1,y1) is
( x 0  x1 )  ( y 0  y 1 )
2
2
(x1,y1)
(x0,y0)
www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
THE LENGTH OF A POLYGONAL PATH?
Add the lengths of the line segments.
www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
THE LENGTH OF A CURVE?
Approximate by chopping it into polygonal pieces
and adding up the lengths of the pieces
www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
APPROXIMATE THE CURVE WITH POLYGONAL
PIECES?
www.spsu.edu/math/Dillon/2254/.../archives/arclength/arclength.ppt
WHAT ARE WE DOING?
 In
essence, we are subdividing
an arc into infinitely many line
segments and calculating the
sum of the lengths of these line
segments.
 For a demonstration, let’s visit
the web.
THE FORMULA:
L

b
a
1   f '  x  dx
2
ARC LENGTH

Note: Many of these integrals cannot be evaluated
with techniques we know. We should use a
calculator to find these integrals.
phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
EXAMPLE PROBLEM

Compute the arc length of the graph of
over [0,1].
f x   x

L 
1
0
3
2
1   f '  x  dx
2
3x
1 
 2
2
1
L 

1
0
2

 dx

3x
1 
 2
2
1
NOW COMES THE FUN PART…
L 

1
0
2

 dx

First, press the Math button and select choice
9:fnInt(
 Next, type the function, followed by X, the lower
bound, and the upper bound.
 Press Enter and you get the decimal approximation
of the integral!

L  fnInt (
1  3 / 2  X

 1 / 2  , X , 0 ,1
L  1 . 44
2

EXAMPLE

Find the arc length of the portion of the curve
2 on the interval [0,1]
y  x

phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
YOU TRY

Find the arc length of the portion of the curve
4
y  x on the interval [0,1]
phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
SURFACE AREA
Compiled by Mrs. King
REVIEW:
Find the volume of the solid created by rotating
3
f ( x )  x about the x-axis on the interval [0,2]



  x  dx
2
0
2
2
3
  x dx
6
0
2
1 7
128 
 x  
7
0
7
Picture from:
http://math12.vln.dreamhosters.com/images/math12.vln.dream
hosters.com/2/2d/Basic_cubic_function_graph.gif
SURFACE AREA OF SOLIDS OF REVOLUTION

When we talk about the surface area of a solid of
revolution, these solids only consist of what is being
revolved.

For example, if the solid was a can of soup, the
surface area would only include the soup can label
(not the top or bottom of the can)
phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
WHAT ARE WE DOING?
Instead
of calculating the
volume of the rotated
surface, we are now going
to calculate the surface
area of the solid of
revolution
THE FORMULA:
L  2

b
a
f  x  1   f '  x  dx
2
EX 2.5

Find the surface area of the surface generated by
4
revolving y  x , 0  x  1, about the x-axis
y'  4 x

S  2

1
x
0

2  fnInt X ^ 4
4
3

1 4x
 dx
3 2
1  4 X ^ 3  , X ,0 ,1
S  3 . 437
phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
2
CLOSURE

Hand in: Find the surface area of the solid created
2
by revolving y  x , 0  x  1 about the xaxis
y' 2x

S  2

1
x
0

2  fnInt X
2
2
1   2 x  dx
2
1  2 X  , X ,0 ,1
S  3 . 81
phs.prs.k12.nj.us/preyes/Calculus%205-4.ppt
2
HOMEWORK
 Page
#

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