Transcript 7 4 Arc Length

7-4: Arc Length
Objectives:
1. To find the arc length
of a smooth curve
Assignment:
• P. 483-486: 3, 5, 6, 9,
15, 16, 19, 21, 23, 25,
26, 33, 34, 65
• Homework Supplement
Warm Up
Find the length of 𝐴𝐵.
Objective 1
You will be able to find the
arc length of a smooth curve
Archimedes’ Method
Recall that when approximating the value of 𝜋,
Archimedes used a series of polygons with straight
sides to get closer to the curved shape of a circle.
We’ll employ a similar method to find the length of
a curve.
PiCircumscribed = 3.46410
Pi = 3.14159
PiInscribed = 3.00000
PiCircumscribed = 3.21539
Pi = 3.14159
PiInscribed = 3.10583
PiCircumscribed = 3.15966
Pi = 3.14159
PiInscribed = 3.13263
Arc Length
Imagine straightening
out the curve shown
and measuring its
length. If you could
feasibly do such a
thing, that
measurement would be
the arc length of the
curve.
Arc Length
To estimate the
arc length of the
curve, we could
approximate the
curve as a series
of line segments.
Arc Length
We could improve our
estimation by taking
smaller and smaller
line segments.
Arc Length
Let’s consider one of
those segments in
the 𝑖 th subinterval.
We could use the
distance formula to
find the length of the
segment.
∆𝑦𝑖
∆𝑥𝑖
𝐿𝑖 =
∆𝑥𝑖
2
+ ∆𝑦𝑖
2
Arc Length
𝑛
𝐿≈
+ ∆𝑦𝑖
∆𝑥𝑖
2
∆𝑥𝑖
2
2
𝑖=1
𝑛
𝐿≈
𝐿≈
𝑖=1
2
+ ∆𝑦𝑖
∆𝑥𝑖
𝑖=1
𝑛
Not quite a Riemann sum
∆𝑥𝑖
∆𝑥𝑖
∆𝑥𝑖
∆𝑥𝑖
∆𝑥𝑖
2
2
∆𝑦𝑖
+
2
∆𝑥𝑖
Since ∆𝑥𝑖 > 0
We could
approximate the
length of the entire
curve by summing 𝑛
such lengths.
2
2 ∆𝑥𝑖
Fancy OneTM
𝑛
𝑛
𝐿≈
∆𝑥𝑖
𝑖=1
2
+ ∆𝑦𝑖
2
∆𝑥𝑖
∆𝑥𝑖
𝐿≈
𝑖=1
∆𝑦𝑖
1+
∆𝑥𝑖
2
∆𝑥𝑖
Now it’s a
Riemann
sum
Arc Length
Taking the limit
as 𝑛 → ∞ yields
the arc length
of the curve.
𝑛
𝐿≈
𝑖=1
𝑛
𝐿 = lim
𝑛→∞
∆𝑦
∆𝑦𝑖
1+
∆𝑥𝑖
𝑖=1
2
∆𝑥𝑖
∆𝑦𝑖
1+
∆𝑥𝑖
𝑑𝑦
As 𝑛 → ∞, → .
∆𝑥
𝑑𝑥
In other words, the slope of
the secant line approaches
the slope of the tangent
line.
𝑏
𝐿=
𝑎
𝑑𝑦
1+
𝑑𝑥
2
𝑑𝑥
2
∆𝑥𝑖
Definition of Arc Length
Let the function given by 𝑦 = 𝑓 𝑥 represent
a smooth curve on the interval 𝑎, 𝑏 . The
arc length 𝑠 of 𝑓 between 𝑎 and 𝑏 is
𝑏
𝑠=
1 + 𝑓′ 𝑥
𝑎
2 𝑑𝑥
𝑦 = 𝑓 𝑥 represents
a smooth curve on
𝑎, 𝑏 if its derivative
is continuous on
𝑎, 𝑏 .
Exercise 1
Find the arc length
of the graph of
𝑦=
𝑥3
6
+
1
2𝑥
on
1
,2
2
.
Exercise 2
Find the arc length
of the graph of
𝑦 − 1 3 = 𝑥 2 on
the interval 0,8 .
Exercise 3
Find the arc
length of the
graph of
𝑦 = ln cos 𝑥
from 𝑥 = 0 to
𝑥 = 𝜋/4.
Exercise 4
Estimate the
length of the
portion of the
hyperbola 𝑥𝑦 =
1 from 1,1 to
1
2, .
2
Exercise 5: AP FRQ
A baker is creating a birthday
cake. The base of the cake is
the region 𝑅 in the first
quadrant under the graph of
𝑦 = 𝑓(𝑥) for 0 ≤ 𝑥 ≤ 30, where
𝜋𝑥
𝑓 𝑥 = 20 sin
. Both 𝑥 and
30
𝑦 are measured in centimeters.
The region 𝑅 is shown in the
figure. The derivative of 𝑓 is
2𝜋
𝜋𝑥
𝑓′ 𝑥 = cos
.
3
30
The region 𝑅 is cut out
of a 30-centimeter-by20-centimeter
rectangular sheet of
cardboard, and the
remaining cardboard
is discarded. Find the
area of the discarded
cardboard.
Exercise 6: AP FRQ
A baker is creating a birthday
cake. The base of the cake is
the region 𝑅 in the first
quadrant under the graph of
𝑦 = 𝑓(𝑥) for 0 ≤ 𝑥 ≤ 30, where
𝜋𝑥
𝑓 𝑥 = 20 sin
. Both 𝑥 and
30
𝑦 are measured in centimeters.
The region 𝑅 is shown in the
figure. The derivative of 𝑓 is
2𝜋
𝜋𝑥
𝑓′ 𝑥 = cos
.
3
30
The cake is a solid with
base 𝑅. Cross sections of
the cake perpendicular to
the 𝑥-axis are semicircles.
If the baker uses 0.05
gram of unsweetened
chocolate for each
cubic centimeter of
cake, how many
grams of
unsweetened
chocolate will be in
the cake?
Exercise 7: AP FRQ
A baker is creating a birthday
cake. The base of the cake is
the region 𝑅 in the first
quadrant under the graph of
𝑦 = 𝑓(𝑥) for 0 ≤ 𝑥 ≤ 30, where
𝜋𝑥
𝑓 𝑥 = 20 sin
. Both 𝑥 and
30
𝑦 are measured in centimeters.
The region 𝑅 is shown in the
figure. The derivative of 𝑓 is
2𝜋
𝜋𝑥
𝑓′ 𝑥 = cos
.
3
30
The cake is a solid with
base 𝑅. Cross sections of
the cake perpendicular to
the 𝑥-axis are semicircles.
Find the perimeter of
the base of the cake.
7-4: Arc Length
Objectives:
1. To find the arc length
of a smooth curve
Assignment:
• P. 483-486: 3, 5, 6, 9,
15, 16, 19, 21, 23, 25,
26, 33, 34, 65
• Homework Supplement