Question 1

A manufacturer wants to create a corrugated metal roofing panel with a sine wave profile given by $$y = \sin\left(\frac{\pi x}{7}\right)$$, where $$x is in $$inches. If the panel is to be 28 inches wide and 2 inches high, what integral gives the width $$w of $$the flat metal sheet needed to make one panel?

Accepted Answer

$$w = \$$int_0$$^{28} \sqrt{1 + \left(\frac{d}{dx} \sin\left(\frac{\pi x}{7}\right)\$$right)^2$$} \, dx$$

Question 2

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

Accepted Answer

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

Question 3

A dam has the shape of a trapezoid with a height of 20 m, a top width of 50 m, and a bottom width of 30 m. If the water level is 4 m from the top, what is the correct setup for the hydrostatic force $$F on $$the dam (using $$\rho$$ for water density and $$g$$ for gravity)?

Accepted Answer

Question 4

Given the law of laminar flow $$v(r) = \frac{P}{4 \eta l}(R^2$ - r^2$)$$, what is the total volume of blood $$F$$ that passes a cross-section per unit time in a vessel of radius $$R$$ and length $$l$$?

Accepted Answer

$$F = \frac{\pi P R^4$}{8 \eta l}$$

Question 5

Which of the following is the correct formula for the area $$A of a $$surface of revolution generated by rotating $$y = f(x)$$, $$a \leq x \leq b$$, about the $$x-$$axis?

Accepted Answer

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

Question 6

A thin horizontal plate of area $$A is $$submerged at depth $$d in a $$fluid of density $$\rho. $$What is the hydrostatic force $$F on $$the plate?

Accepted Answer

$$F = \rho g A d$$

Question 7

Suppose $$y = f(x) is $$rotated about the $$y-$$axis from $$x = a to$$ $$x = b. $$Which method is most appropriate for finding the volume of the resulting solid?

Accepted Answer

Shell method

Question 8

Given the differential equation $$\frac{dP}{dt} = kP$$, which of the following best describes the population $$P$$ over time if $$k \u003e 0$$?

Accepted Answer

$$P$$ grows exponentially

Question 9

If $$y = e$$^{2t}$$ is a solution to $$\frac{$$d^2y$$}{$$dt^2$$} - 5 \frac{dy}{dt} + k y = 0$$, what is the value of k$?

Accepted Answer

$$k = 6$$

Question 10

Which trigonometric substitution is appropriate for evaluating $$\int \sqrt{9 - x^2$} \, dx$$?

Accepted Answer

$$x = 3 \sin 	heta$$

Question 11

A population $$P$$ grows according to $$\frac{dP}{dt} = 0.1P. If$$ $$P(0) = 100$$, what is $$P(10)$$?

Accepted Answer

$$P(10) = 100e$$^{1}$$

Question 12

Which of the following integrals represents the arc length of the curve $$x = g(y)$$ from $$y = c to$$ $$y = d$$?

Accepted Answer

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

Question 13

A blood vessel has radius $$R$$ and length $$l. $$According to the law of laminar flow, what happens to the total blood flow $$F if $$the radius is doubled, all else held constant?

Accepted Answer

$$F$$ increases by a factor of 16

Question 14

Which of the following best describes the process of setting up an integral to find the work required to pump water out of a tank?

Accepted Answer

Divide the tank into thin slices, find the force needed to move each slice to the top, and integrate over the height of the tank.

Question 15

Given the differential equation $$y' = \frac{1}{2}(y^2$ - 1)$$ with initial condition $$y(0) = 2$$, which of the following is a solution?

Accepted Answer

$$y = \frac{1 + ce$$^{t}$$}{1 - ce$$^{t}$$}$$ for some $$c$$

Question 16

Which of the following is a correct interpretation of the definite integral $$\int_a^b F(x) \, dx in $$the context of hydrostatic force on a vertical plate submerged in a fluid?

Accepted Answer

It represents the total force exerted by the fluid on the plate between depths $$a$$ and $$b.$$

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