Q: A 5-kg bucket is lifted 20 m with a rope weighing 0.08 kg/m. What is the total work done lifting the bucket and rope (using $$g = 9.8 m/s$^2$$)?

$$980 + \$$int_0$$^{20} 0.08(20-x) \times 9.8\,dx$$

Q: The force exerted by a fluid against one side of a submerged vertical plate from $$y = a to$$ $$y = b is$$:

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

Question 1

A solid has cross-sectional area $$A(x)$$ perpendicular to the x-axis from $$x = a to$$ $$x = b. $$Which integral gives the volume of the solid?

Accepted Answer

$$\int_a^b A(x)\,dx$$

Question 2

A pyramid is 3 m high with a square base 3 m on a side. The cross-section perpendicular to the altitude $$x m $$down from the vertex is a square $$x m on a $$side. What is the volume of the pyramid?

Accepted Answer

$$\$$int_0$$^3 x^2$\,dx = 9$$

Question 3

Cavalieri’s principle states that two solids with equal altitudes and identical cross-sectional areas at each height have:

Accepted Answer

The same volume

Question 4

Find the volume of the solid generated by revolving the region between $$y = \sqrt{x}$$ and the x-axis from $$x = 0 to$$ $$x = 4$$ about the x-axis.

Accepted Answer

$$\$$int_0$$^4 \pi (\sqrt{x})^2$\,dx$$

Question 5

When using the washer method to find the volume of a solid of revolution, the formula is:

Accepted Answer

$$\int_a^b \pi (R(x)^2$ - r(x)^2$)\,dx$$

Question 6

The shell method for finding the volume of a solid generated by revolving a region about a vertical line $$x = L is $$given by:

Accepted Answer

$$\int_a^b 2\pi (x - L) f(x)\,dx$$

Question 7

The arc length of the curve $$y = f(x)$$ from $$x = a to$$ $$x = b is$$:

Accepted Answer

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

Question 8

If $$x = g(y)$$ and $$g' is $$continuous on $$[c, d]$$, the arc length from $$y = c to$$ $$y = d is$$:

Accepted Answer

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

Question 9

The surface area generated by revolving $$y = f(x)$$ about the x-axis from $$x = a to$$ $$x = b is$$:

Accepted Answer

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

Question 10

The work $$W$$ done by a constant force $$F$$ moving an object a distance $$d is$$:

Accepted Answer

$$W = Fd$$

Question 11

According to Hooke’s Law, the force required to compress a spring by $$x$$ units is:

Accepted Answer

$$F = kx$$

Question 12

A 5-kg bucket is lifted 20 m with a rope weighing 0.08 kg/m. What is the total work done lifting the bucket and rope (using $$g = 9.8 m/s$^2$$)?

Accepted Answer

$$980 + \$$int_0$$^{20} 0.08(20-x) 	imes 9.8\,dx$$

Question 13

The pressure at depth $$h in a $$fluid of weight-density $$w is$$:

Accepted Answer

$$p = wh$$

Question 14

The force exerted by a fluid against one side of a submerged vertical plate from $$y = a to$$ $$y = b is$$:

Accepted Answer

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

Question 15

The moment about the origin for masses $$m_1$$, $$m_2$$, $$m_3 at $$positions $$x_1$$, $$x_2$$, $$x_3 is$$:

Accepted Answer

$$m_1x_1 + m_2x_2 + m_3x_3$$

Question 16

For a thin plate of constant density covering a region between $$y = g(x)$$ and $$y = f(x)$$, the mass is:

Accepted Answer

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

Question 17

The centroid of a semicircular wire of radius $$a$$ lies at:

Accepted Answer

$$(0, \frac{2a}{\pi})$$

Question 18

Pappus’s Theorem for volumes states that the volume generated by revolving a plane region about an axis is:

Accepted Answer

Area of the region times the distance traveled by the centroid

Question 19

Using Pappus’s Theorem, the volume of a torus generated by revolving a disk of radius $$a$$ about an axis at distance $$b$$ from its center is:

Accepted Answer

$$2\pi^2$ a^2$ b$$

Chapter 6: Applications of Definite Integrals – Study Guide

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