Question 1

Which of the following integrals correctly represents the arc length of the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 2

Given the curve $$y = x^2$ from x = 0$ to x = 1$, which of the following is the correct setup for the arc length?

Accepted Answer

$$\$$int_0$$^1 \sqrt{1 + (2x)^2$} \, dx$$

Question 3

Suppose you want to find the arc length of $$x = g(y)$$ from $$y = c to$$ $$y = d. $$Which formula should you use?

Accepted Answer

$$\int_c^d \sqrt{1 + [g'(y)]^2} \, dy$$

Question 4

A cable hangs in the shape of $$y = \cosh(x)$$ between $$x = a$$ and $$x = b. $$What is the length of the cable between these points?

Accepted Answer

$$\sinh(b) - \sinh(a)$$

Question 5

Which of the following best describes the process for finding the arc length of a curve $$y = f(x)$$?

Accepted Answer

Graph the segment, differentiate $$f(x)$$, set up the integral $$\int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$, and evaluate.

Question 6

Find the arc length of the straight line segment connecting $$(1, 3)$$ and $$(3, 7).$$

Accepted Answer

$$2\sqrt{5}$$

Question 7

For the curve $$y = (1 - x^2$)^{1/2}$$ from $$x = -1 to$$ $$x = 1$, what is the arc length?

Accepted Answer

$$\pi$$

Question 8

Which of the following is a real-world application of arc length calculation?

Accepted Answer

Determining the length of a cable hanging between two poles.

Question 9

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

Accepted Answer

$$b - a$$

Question 10

Which function, besides $$\cosh(x)$$, has the property that the area under the curve from $$a to$$ $$b$$ equals the arc length over the same interval?

Accepted Answer

$$y = e^x ($$since $$\int_a^b e^x dx = e^b - e^a$$ and arc length is also $$e^b - e^a$$ for this function)

Question 11

Given $$y = \frac{2}{3}x$$^{3/2}$$ + 2$$ from $$x = 0 to$$ $$x = 1$$, what is the correct integral for the arc length?

Accepted Answer

$$\$$int_0$$^1 \sqrt{1 + x} \, dx$$

Question 12

If $$y^3$$ = $$x^2$$, and you want to find the arc length from $$x = 0 to$$ $$x = 1$$, which substitution is most helpful?

Accepted Answer

Solve for $$x in $$terms of $$y$$ and use $$x = y$$^{3/2}$$

Question 13

Which of the following best explains why the arc length formula involves a square root?

Accepted Answer

It comes from the Pythagorean theorem applied to small segments of the curve.

Question 14

A road is designed following the curve $$y = f(x)$$ for $$x in$$ $$[a, b]. $$Why is arc length important in this context?

Accepted Answer

It gives the actual distance a car would travel along the road.

Question 15

If $$f(x) = 2x + 1$$, what is the arc length from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$2\sqrt{5}$$

Arc Length: Calculus Applications and Techniques

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