8.4.1How long is the curve?

Marga’s little sister is making a poster for her algebra class and she wants to use a piece of yarn to show a function on a graph. The function is $y = x^2$ for a domain of $–4 ≤ x ≤ 4$. She asks Marga to help her figure out how much yarn she will need. Marga realizes that finding the length of a curve is something she has never done, but it seems to be a similar situation to others that she has studied in calculus.

Using the Lesson 8.4.1 Resource Page work with your team to determine how much yarn Marga’s sister will need as accurately as possible. The length of a curve is called the arc length of the curve.

SLOPE AND THE LENGTH OF A SECANT SEGMENT

Sketch a graph of any smooth continuous function, $f$. Then draw a segment connects two arbitrary points on the graph at $x = a$ and at $x = a + h$.

1. Write an equation for $m$, the slope of the segment.

2. Write an equation for $l$, the length of the segment.

3. How accurately do you think $m$ approximates the slope at $x = a$? How accurately do you think $l$ approximates the arc length between $x = a$ and $x = a + h$? How can you improve these approximations?

SLOPE AND LENGTH OF AN INFINITESIMALLY SMALL SECANT SEGMENT

1. Let $h → 0$ and use a limit to write a new expression for $m$, the slope of the infinitely small segment. What does this equation represent, in terms of the function $f$ ?

2. As $h → 0$, $l$ becomes infinitesimally small. Use a limit to write a new expression for $dl$, the length of the infinitesimally small segment from $x = a$ to $x = a + h$. Do not simplify yet.

3. Using your equation from part (a), solve for $f(a + h) – f(a)$. Then substitute that expression into your equation from part (b). Do not simplify yet.

4. On an $xy$-coordinate plane, $h = ∆x$ and as $h → 0$, $∆x → dx$. Substitute $dx$ into the equation in part (c).  

5. The limit in part (d) can be used to calculate the arc length of an infinitesimally small piece of the graph of $f$ at $x = a$. Adjust the argument in part (d) to find the infinitesimally small arc length at:

   1. $x = b$ 

   2. $x = 10$ 

   3. at any value of $x$  

CALCULATING THE ARC LENGTH OVER A GIVEN INTERVAL

Remember Margo’s little sister, who wanted to know the exact length of the arc on the graph of $y = x^2$ over the domain $–4 ≤ x ≤ 4$? Help Margo use calculus to measure the length of this arc.

1. Start by writing a definite integral that represents the general arc length of a smooth continuous function, $f$, over the interval $a ≤ x ≤ b$. Use algebra to show the $dx$ in the integral. Be prepared to share your answer with the class.

2. Compute the arc length for the graph of $y = x^2$ on $–4 ≤ x ≤ 4$, which Margo’s sister is preparing on inch grid paper.

3. Explain why the arc length formula in part (a) requires that $f$ is a “smooth and continuous” function?

Express the arc length of each of the following curves as an integral, then evaluate:

1. $f (x) = \sin(x)$, from $x = 0$ to $x = π$. 

2.  $g(x) =\frac { 2 } { 3 }(x − 1)^{3/2}$, from $x = 1$ to $x = 9$. 

Express the arc length of each of the following curves as an integral, then evaluate: Homework Help ✎

1. $f (x) = e^x$, from $x = -2$ to $x = 2$

2. $f (x) = \ln(x)$, from $x = 1$ to $x = e$

Calculate the volume of the solid constructed as follows: The base of the solid is the region is formed by the curve $y = e^{x^2}$ and the line $y = e$, while the cross-sections perpendicular to the $y$-axis are squares. Homework Help ✎

Without looking at the graph, use Newton’s Method to approximate the roots of $f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$ if $x_1 = 1$. What happened? Homework Help ✎

The region $M$ is enclosed by the functions $f(x) = x^2 + 1$ and $g(x) = 2x + 1$. Set up, but do not evaluate, the integrals to represent the volumes of the solids formed by revolving $M$ about each of the following axes. Homework Help ✎

Examine the integrals below. Consider the multiple tools available for evaluating integrals and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int _ { 3 } ^ { 8 } H ( t ) d t = - 1.25$ where $H$ represents the rate that snow accumulates on a certain day in January. $H$ is measured in feet per hour and $t = 0$ represents 12:00 p.m. Homework Help ✎

1. Write a complete description about what the integral is computing. Use correct units and be sure to mention the meaning of the bounds in your description.

2. Compute the value of $\frac { 1 } { 5 } \int _ { 3 } ^ { 8 } H ( t ) d t$ and interpret its meaning in the context of this situation.

1. What was the average velocity of the starship during this interval?

2. At what time, if any, was the ship’s velocity equal to its average velocity? Does the Mean Value Theorem apply in this case? Explain.

3. What was the average acceleration of the starship during this interval?

4. At what time, if any, was the ship’s acceleration equal to this average acceleration? Does the Mean Value Theorem apply in this case? Explain.