Q: A vertical plate is submerged in a fluid of weight-density $$w$$ from $$y = a to$$ $$y = b. If$$ $$L(y) is $$the width at depth $$y$$, which integral gives the total fluid force on the plate?

$$F = \int_a^b w \cdot (\text{depth}) \cdot L(y) \, dy$$

Q: A rectangular window is 63 in. wide and runs from 0.5 in. to 33.5 in. below the water surface. If the weight-density of seawater is 64 lb/ft$$^3$$, what is the correct integral for the fluid force on the window?

$$F = \int_{0.5/12}^{33.5/12} 64 \cdot (\text{depth}) \cdot 63/12 \, dy$$

Question 1

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

Accepted Answer

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

Question 2

Given the curve $$x^{2/3}$$ + $$y^{2/3} = 1$, what is the correct integral to find the length of the first-quadrant portion between x$$ = \sqrt{2}/2$$ and x = 1$?

Accepted Answer

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

Question 3

Which of the following represents the surface area $$S of $$the surface generated by revolving $$y = f(x)$$, $$a \leq x \leq b$$, about the $$x-$$axis?

Accepted Answer

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

Question 4

A tank is 20 ft tall, 12 ft wide, and 10 ft deep, filled with water (weight-density $$w). $$What is the correct integral for the work required to pump all the water to the top?

Accepted Answer

$$W = \$$int_0$$^{20} w \cdot 12 \cdot 10 \cdot (20 - y) \, dy$$

Question 5

A vertical plate is submerged in a fluid of weight-density $$w$$ from $$y = a to$$ $$y = b. If$$ $$L(y) is $$the width at depth $$y$$, which integral gives the total fluid force on the plate?

Accepted Answer

$$F = \int_a^b w \cdot (	ext{depth}) \cdot L(y) \, dy$$

Question 6

A triangular plate with base 6 ft and height 5 ft is submerged vertically with its vertex at the bottom and base at the surface. What is the width function $$L(y) at $$depth $$y$$?

Accepted Answer

$$L(y) = \frac{6}{5}y$$

Question 7

Given a lamina occupying the region between $$y = f(x)$$ and $$y = g(x)$$, $$a \leq x \leq b$$, with constant density $$\delta$$, what is the formula for the $$x-$$coordinate of the centroid?

Accepted Answer

$$\bar{x} = \frac{1}{M} \int_a^b \delta x [f(x) - g(x)] \, dx$$

Question 8

A rod of negligible mass has two masses $$m_1$$ and $$m_2 at $$distances $$d_1$$ and $$d_2$$ from the fulcrum. The rod balances when:

Accepted Answer

$$m_1$$ $$d_1$$ = $$m_2$$ $$d_2$$

Question 9

Which of the following integrals gives the moment about the $$y-$$axis for a lamina with density $$\delta$$ occupying $$y = f(x)$$, $$a \leq x \leq b$$?

Accepted Answer

$$M_y = \int_a^b \delta x f(x) \, dx$$

Question 10

A cone is generated by revolving the line $$y = x/2$$, $$0 \leq x \leq 4$$, about the $$y-$$axis. What is the lateral surface area using calculus?

Accepted Answer

$$S = 2\pi \$$int_0$$^2 y \sqrt{1 + \left(\frac{dx}{dy}\$$right)^2$$} \, dy$$

Question 11

A rectangular window is 63 in. wide and runs from 0.5 in. to 33.5 in. below the water surface. If the weight-density of seawater is 64 lb/ft$$^3$$, what is the correct integral for the fluid force on the window?

Accepted Answer

$$F = \int_{0.5/12}^{33.5/12} 64 \cdot (	ext{depth}) \cdot 63/12 \, dy$$

Question 12

Which of the following best describes the physical meaning of the centroid $$(\bar{x}, \bar{y}) of a $$lamina?

Accepted Answer

It is the point where the lamina would balance if supported at that point.

Question 13

If a plate is submerged vertically in a fluid and the top of the plate is at $$y = a$$, what is the expression for the depth of a strip at $$y$$?

Accepted Answer

$$(a - y)$$

Question 14

For a region between $$y = f(x)$$ and $$y = g(x)$$, $$a \leq x \leq b$$, with constant density $$\delta$$, what is the formula for the total mass $$M$$?

Accepted Answer

$$M = \int_a^b \delta [f(x) - g(x)] \, dx$$

Question 15

Which of the following is the correct formula for the $$y-$$coordinate of the centroid for a region between $$y = f(x)$$ and $$y = g(x)$$, $$a \leq x \leq b$$, with constant density $$\delta$$?

Accepted Answer

$$\bar{y} = \frac{1}{M} \int_a^b \frac{\delta}{2} \left([f(x)]^2 - [g(x)]^2\right) \, dx$$

Calculus Study Guide: Arc Length, Surface Area, Fluid Forces, and Center of Mass

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