Calculus 8.13 Arc Length Notes

Calculus 8.13 Arc Length

Notes

Write your questions and thoughts here!

The idea behind finding arc length is very similar to the way we find area using calculus. We are going to divide the curve into a large quantity of small segments, find their lengths and then add them up.

Recall the distance formula:

We can approximate the length of the graph of a function by using line segments whose endpoints are partitioned between , . Think of it this way: and where .

y

x

Arc Length

If a function represents a smooth continuous curve on the closed interval , , the arc length of between and is given by

1

1. Find the arc length of the graph

8 from 1 to 4.

Write your questions Often the problem will only ask that you set up the integral, but we could take this further and thoughts here! and use the calculator

2. Set up an integral that represents the length of the curve sin , for 0 and use a calculator to find the value.

Proof: 1.

2.

3.

4.

5.

1

6.

1

7. The approximation improves as we take the

number of segments to infinity.

lim

1

8. Because exists for every in the interval

, the MVT guarantees that there is a

value in the interval such that

9.

10.

lim

1

11. Using the definition of integration, we get

1

8.13 Arc Length

Calculus

1. Find an expression for the length of the curve sin from 0 to

Practice

. Do Not Evaluate.

2. The length of a curve from 1 to 3 is given by 1 4 . If the point 1, 6 is on the curve, which of the following could be an equation for this curve? A. 1

B. 4 1 C. 5 D. 6 E. 1

3. Calculator active. Suppose

sin , for 0 . What is the length of the arc along the

curve for 0 to /7.

4. No Calculator. Let 3 and be an antiderivative of . a. Find b. Find an expression for the length of the graph of from to .

c. If 0 and 8, find the length of the graph of from to .

5. Calculator active. Consider the region bounded by the graphs of 4 and 5. a. Write an expression using one or more integrals that could be used to find the perimeter of this region.

b. Find the perimeter.

6. Find an integral that gives the length of the graph cos between and , where 0 .

7. Calculator active. Let be a function with derivative 1. What is the length of the graph of from 0 to 2.5?

8. Find an integral that is equal to the length of the curve 2, 5 .

from the point 0, 0.143 to the point

9. Find an expression for the length of the graph of between 1 and 3.

10. Calculator active. The trajectory of a ball thrown from a height of 160 meters is given by the equation

160 until it hits the water where is the height of the ball above the water and is the horizontal

distance traveled in meters. Find the distance traveled by the ball from the time it is thrown until it hits the water.

8.13 Arc Length

11. Which of the following integrals gives the length of the curve

Test Prep

from 1 to 3?

A.

1

B.

1

C.

4 9

D.

1

12. Calculator active. What is the length of the curve 1 sin from 0 to 4?

13.

y

x

1

3 4

for 0 4

5

1 4

4

6

for 4 8 for 8 10

A mountain hike consists of a steady incline followed by a curved hill and then a flat valley. The mountain hike is modeled by the piecewise-defined function above, and the graph of is shown in the figure above. Which of the following expressions gives the total length of the hike from 0 to 10.

A. 2

1

B. 2

1

6 1 6

C. 7

1 1 6

D. 7

1 6

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