Q: A plate is submerged vertically 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 one side of the plate?

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

Question 1

Given the curve $$y = (1 - x$$^{2/3}$$)^{3/2}$$ for $$\sqrt{2}/2 \leq x \leq 1$, what is the correct integral setup to find the length of the arc in the first quadrant, as described in the astroid example?

Accepted Answer

$$\int_{\sqrt{2}/2}^1 \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} \, dx$$

Question 2

Which formula gives 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 3

Suppose you revolve the curve $$y = f(x)$$, $$a \leq x \leq b$$, about the $$x-$$axis. Which integral gives the surface area $$S of $$the resulting surface of revolution?

Accepted Answer

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

Question 4

A vertical cylindrical tank is 30 ft high and 20 ft in diameter, full of kerosene (weight-density 51.2 lb/ft³). What is the correct integral to compute the work required to pump all the kerosene to the top of the tank?

Accepted Answer

$$W = \$$int_0$$^{30} 51.2 \cdot \pi \cdot 10^2$ \cdot (30 - y) \, dy$$

Question 5

A plate is submerged vertically 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 one side of 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. If the weight-density of the fluid is $$w$$, what is the correct expression for the fluid force on one side of the plate?

Accepted Answer

$$F = w \$$int_0$$^5 y \cdot \left(\frac{6}{5}y\right) \, dy$$

Question 7

The centroid $$(\bar{x}, \bar{y}) of a $$thin plate of constant density $$\delta$$ bounded above by $$y = f(x)$$, below by $$y = g(x)$$, and between $$x = a$$ and $$x = b is $$given by:

Accepted Answer

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

Question 8

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 9

A rectangular aquarium 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/ft³, what is the correct integral for the fluid force on the window (convert all units to feet)?

Accepted Answer

$$F = \int_{0.0417}^{2.792} 64 \cdot y \cdot 5.25 \, dy$$

Question 10

Which of the following best describes the physical meaning of the moment about the $$y-$$axis for a lamina of density $$\delta$$ occupying region $$R$$?

Accepted Answer

It is the sum (or integral) of $$x$$ times the mass element, representing the tendency of the mass to rotate about the $$y-$$axis.

Question 11

If a curve is given as $$x = g(y)$$, $$c \leq y \leq d$$, what is the formula for the arc length?

Accepted Answer

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

Question 12

A lamina occupies the region between $$y = x^2$ and y = 4$ for 0$$ \leq x \leq 2$$, with constant density $$\delta$$. What is the total mass M$?

Accepted Answer

$$M = \delta \$$int_0$$^2 [4 - x^2$] \, dx$$

Question 13

Which of the following is the correct formula for the surface area of the surface generated by revolving $$x = g(y)$$, $$c \leq y \leq d$$, about the $$y-$$axis?

Accepted Answer

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

Question 14

A thin plate of constant density $$\delta$$ occupies the region between $$x = 0$$ and $$x = 1$$, above the $$x-$$axis and below $$y = x^2$. What is the y$-coordinate of the centroid?

Accepted Answer

$$\bar{y} = \frac{\$$int_0$$^1 \frac{\delta}{2} [x^4$] \, dx}{\$$int_0$$^1 \delta x^2$ \, dx}$$

Question 15

Which of the following best describes the physical meaning of the centroid of a region?

Accepted Answer

It is the balance point of the region if it were made of a thin, uniform plate.

Arc Length, Surface Area of Revolution, and Fluid Force Problems – Step-by-Step Calculus Guidance

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