Transcript Day 5: Arc Length and Surface Area

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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