Question 1

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

Accepted Answer

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

Question 2

Find the exact length of the curve $$y = 3 + 2x$$^{3/2}$$ for 0$$ \leq x \leq 1$$.

Accepted Answer

$$L = \$$int_0$$^1 \sqrt{1 + 9x}$$ \, dx = \frac{2}{27}[(1 + $$9x)^{3/2}$$]_$$0^1$

Question 3

A straight line $$y = 4x - 5 is $$given for $$-1 \leq x \leq 2. $$What is the length of this curve?

Accepted Answer

$$L = \int_{-1}^2 \sqrt{1 + 16} \, dx = 3\sqrt{17}$$

Question 4

Given $$x = y^4$ + \frac{1}{4y^2$}$$ for $$1 \leq y \leq 2$$, which integral represents the arc length of the curve?

Accepted Answer

$$L = \$$int_1$$^2 \sqrt{1 + \left(\frac{dx}{dy}\$$right)^2$$} \, dy$$

Question 5

Suppose $$x = \frac{1}{3}\sqrt{y}(y-3)$$ for $$9 \leq y \leq 25. $$What is $$\frac{dx}{dy}$$?

Accepted Answer

$$\frac{dx}{dy} = \frac{1}{3} \left( \frac{1}{2}y$$^{-1/2}$$(y-3) + \sqrt{y} \right)$$

Question 6

Let $$y = \$$int_1$$^x \sqrt{t^3$ - 1} \, dt$$ for $$4 \leq x \leq 9. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \sqrt{x^3$ - 1}$$

Question 7

Find the exact area of the surface obtained by rotating $$y = x^3$ for 0$$ \leq x \leq 5$$ about the x$-axis.

Accepted Answer

$$S = 2\pi \$$int_0$$^5 x^3$ \sqrt{1 + 9x^4$} \, dx$$

Question 8

Which formula gives the surface area $$S$$ when the curve $$y = f(x)$$, $$a \leq x \leq b$$, is rotated about the $$x-$$axis?

Accepted Answer

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

Question 9

Find the surface area generated by rotating $$y = \frac{x^3$}{6} + \frac{1}{2x}$$, $$\frac{1}{2} \leq x \leq 1$$, about the $$x-$$axis.

Accepted Answer

$$S = 2\pi \int_{1/2}^1 \left( \frac{x^3$}{6} + \frac{1}{2x} \right) \sqrt{1 + \left( \frac{x^2$}{2} - \frac{1}{2x^2$} \$$right)^2$$} \, dx$$

Question 10

A parabolic arch $$y = 4 - x^2$ for 0$$ \leq x \leq \frac{5}{3}$$ is rotated about the y$-axis. Which integral gives the surface area?

Accepted Answer

$$S = 2\pi \$$int_0$$^{5/3} x \sqrt{1 + [f'(x)]^2} \, dx$$, where $$f(x) = 4 - x^2$

Question 11

A cable is suspended in the shape of $$y = 3 + 2x$$^{3/2}$$ from x = 0$ to x = 1$. If the cable weighs 5 N per meter, what is the total weight of the cable?

Accepted Answer

$$5 	imes \$$int_0$$^1 \sqrt{1 + 9x} \, dx$$

Question 12

Suppose you want to find the length of the curve $$y = \$$int_1$$^x \sqrt{t^3$ - 1} \, dt$$ for $$4 \leq x \leq 9. $$Which integral represents the arc length?

Accepted Answer

$$L = \$$int_4$$^9 \sqrt{1 + [\sqrt{x^3$ - 1}]^2} \, dx$$

Question 13

Which of the following best describes the process for finding the surface area generated by rotating $$x = g(y)$$, $$c \leq y \leq d$$, about the $$y-$$axis?

Accepted Answer

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

Question 14

A water pipe is shaped as $$y = x^3$ for 0$$ \leq x \leq 5$$ and is rotated about the x$-axis. If the pipe is to be painted on the outside, which integral gives the area to be painted?

Accepted Answer

$$2\pi \$$int_0$$^5 x^3$ \sqrt{1 + 9x^4$} \, dx$$

Question 15

If $$y = 4x - 5$$ for $$-1 \leq x \leq 2$$, what is the geometric interpretation of the arc length over this interval?

Accepted Answer

It is the straight-line distance between the points $$(-1, -9)$$ and $$(2, 3).$$

Question 16

Which of the following is a correct step in finding the arc length of $$x = y^4$ + \frac{1}{4y^2$}$$ for $$1 \leq y \leq 2$$?

Accepted Answer

Compute $$\frac{dx}{dy} = 4y^3$ - \frac{1}{2y^3$}$$

Question 17

A curve $$y = f(x) is $$rotated about the $$y-$$axis. Which formula gives the surface area?

Accepted Answer

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

Arc Length and Surface Area Guidance for Calculus (Sections 6.3 & 6.4)

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