Q: Which of the following is the correct formula for the volume of a solid of revolution using the washer method?

$V = \pi \int_{a}^{b} \left[R_{\text{outer}}^2 - R_{\text{inner}}^2\right] \, dx$

Question 1

Given the curve $y = \frac{3}{5}x^{5/6}$ for $1 \leq x \leq 3$, what is the exact length of the curve?

Accepted Answer

$\int_{1}^{3} \sqrt{1 + \left(\frac{d}{dx} \left(\frac{3}{5}x^{5/6}\right)\right)^2} \, dx$

Question 2

For the region bounded by $y = 6 - x^2$ and $y = 2$, which of the following integrals gives the volume when rotated about the x-axis?

Accepted Answer

$\pi \int_{-2}^{2} \left[(6 - x^2)^2 - 2^2\right] \, dx$

Question 3

A 150 lb cable is 25 ft long and hangs vertically from the top of a tall building. How much work is required to lift the entire cable to the top?

Accepted Answer

$\int_{0}^{25} 150x \, dx$

Question 4

How much work is required to lift the cable only up to 15 ft from the top (i.e., from $x = 10$ to $x = 25$)?

Accepted Answer

$\int_{10}^{25} 150x \, dx$

Question 5

Which of the following best describes the method to find the arc length of $y = f(x)$ from $x = a$ to $x = b$?

Accepted Answer

$\int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx$

Question 6

If a region is bounded by $y = 6 - x^2$ and $y = 2$, what are the x-values where these curves intersect?

Accepted Answer

$x = \pm 2$

Question 7

Which integral represents the work required to lift a cable of weight density $w$ and length $L$ hanging from a building?

Accepted Answer

$\int_{0}^{L} w(L - x) \, dx$

Question 8

A cable weighs 150 lb and is 25 ft long. What is the average force required to lift the cable to the top of the building?

Accepted Answer

$75$ lb

Question 9

Which of the following is the correct setup for the volume of the solid formed by rotating the region between $y = 6 - x^2$ and $y = 2$ about the y-axis?

Accepted Answer

$2\pi \int_{2}^{6} (\sqrt{6 - y} - \sqrt{y - 2})y \, dy$

Question 10

If the work required to lift a cable is given by $W = \int_{0}^{L} w(L - x) \, dx$, what does the expression $L - x$ represent?

Accepted Answer

The distance each segment of the cable must be lifted

Question 11

Which of the following is a correct interpretation of the definite integral $\int_{a}^{b} f(x) \, dx$ in the context of work?

Accepted Answer

The total work done by a variable force $f(x)$ over the interval $[a, b]$

Question 12

If $y = \frac{3}{5}x^{5/6}$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{3}{5} \cdot \frac{5}{6} x^{-1/6}$

Question 13

Which of the following is the correct formula for the volume of a solid of revolution using the washer method?

Accepted Answer

$V = \pi \int_{a}^{b} \left[R_{	ext{outer}}^2 - R_{	ext{inner}}^2ight] \, dx$

Question 14

A cable of length $L$ and weight density $w$ is lifted at a constant rate. What is the total work done?

Accepted Answer

$\frac{1}{2}wL^2$

Applications of Integration: Arc Length, Volume, and Work

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