Q: Which of the following is a correct step in finding the arc length of $$y = r^2$ - x^2$ from x = -r$ to x = r$?

Use the substitution $$x = r \sin \theta to $$simplify the integral.

Question 1

Which of the following is the correct formula for the arc length $$L of a $$smooth curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{1 + (f'(x))^2$} \, dx$$

Question 2

Given the curve $$y = x$$^{3/2}$$, what is the value of f$$'(x)$$?

Accepted Answer

$$f'(x) = \frac{3}{2} x$$^{1/2}$$

Question 3

What is the arc length of the curve $$y = x$$^{3/2}$$ from x = 0$ to x = 4$?

Accepted Answer

$$L = \frac{8}{3} (10$$^{3/2}$$ - 1)$$

Question 4

Which substitution is most useful for evaluating $$\int \sqrt{r^2$ - x^2$} \, dx$$?

Accepted Answer

$$x = r \sin 	heta$$

Question 5

The circumference of a circle of radius $$r$$ can be found using the arc length formula. What is the result?

Accepted Answer

$$2\pi r$$

Question 6

For the curve $$y = 2e^x + e$$^{-x}$$ on $$[0, \ln 2]$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 2e^x - e$$^{-x}$$

Question 7

Which of the following best describes the geometric meaning of the arc length integral $$\int_a^b \sqrt{1 + (f'(x))^2$} \, dx$$?

Accepted Answer

It sums the lengths of infinitesimal line segments along the curve.

Question 8

Suppose you want to approximate the length of a curve between $$x = a$$ and $$x = b$$ using $$n$$ line segments. What is the expression for the length of one segment between $$x_{i-1}$$ and $$x_i$$?

Accepted Answer

$$\sqrt{(x_i - x_{i-1})^2$ + (f(x_i) - f(x_{i-1}))^2$}$$

Question 9

If $$f'(x) is $$always zero on $$[a, b]$$, what is the arc length of $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$b - a$$

Question 10

A cable is suspended between two poles 10 meters apart, and its height above the ground at position $$x is $$given by $$y = 2 + 0.1x^2$ for x$ in $$[-5, 5]$$. What integral gives the length of the cable?

Accepted Answer

$$\int_{-5}^{5} \sqrt{1 + (0.2x)^2$} \, dx$$

Question 11

Which of the following is a correct step in finding the arc length of $$y = r^2$ - x^2$ from x = -r$ to x = r$?

Accepted Answer

Use the substitution $$x = r \sin 	heta to $$simplify the integral.

Question 12

A road follows the path $$y = x$$^{3/2}$$ from x = 0$ to x = 4$. If a car travels along this road, what is the total distance traveled?

Accepted Answer

$$\frac{8}{3} (10$$^{3/2}$$ - 1)$$

Question 13

Which of the following is NOT a valid application of the arc length formula?

Accepted Answer

Finding the area under a curve.

Question 14

If $$y = mx + b is a $$straight line, what is the arc length from $$x = x_0$ to x$$ = $$x_1$$?

Accepted Answer

$$\sqrt{(x_1$ - x_0$)^2$$ + ($$y_1$$ - $$y_0)^2$}$$

Question 15

Which of the following best explains why the arc length formula involves $$\sqrt{1 + (f'(x))^2$}$$?

Accepted Answer

It comes from the Pythagorean theorem applied to infinitesimal changes in $$x$$ and $$y.$$

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