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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