Question 1

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of  . 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Suppose you make napkin rings by drilling...

Question 2

Find the derivative of the function 
A) 
B) 
C) 
D)

Accepted Answer

Since the function given is in the form of a hexadecimal number, we can assume that it is a constant function. The derivative of any constant function is 0. However, it is notable that the given function is slightly different from the options provided. Among the options, only option B differs by only one digit from the given function, so the best option is to choose B as the answer.

Question 3

A tank has the shape of an inverted right circular cone with a base radius of 4 m and a height of 9 m. If the tank is filled to a height of 3 m, find the work required (to the nearest joule) to empty the tank by pumping the water over the top of the tank. (The mass of water is 1000 kg/m³ and the force of gravity is 9.8 m/sec².)

Accepted Answer

The answer of A tank has the shape of an...

Question 4

In a steam engine the pressure and volume of steam satisfy the equation  , where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work done by the engine (in ft-lb) during a cycle when the steam starts at a pressure of  and a volume of  and expands to a volume of  Use the fact that the work done by the gas when the volume expands from  to volume  is  .

Accepted Answer

The answer of In a steam engine the pressure and...

Question 5

An aquarium  m long,  m wide, and 1 m deep is full of water. Find the work (in J) needed to pump half of the water out of the aquarium. (Use the facts that the density of water is  and  )

Accepted Answer

The answer of An aquarium  m long,  m...

Question 6

Use cylindrical shells to find the volume of the solid. A sphere of radius  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use cylindrical shells to find the volume...

Question 7

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.  y-axis
A)  $\pi$
B)  $\pi$
C)  $\pi$
D)  $\pi$

Accepted Answer

The answer of Use the method of cylindrical shells to...

Question 8

Find the derivative of the function. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the derivative of the function. 
A)...

Question 9

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b]. 
A) 
B)- 
C)- 
D)

Accepted Answer

We cannot determine the exact function and interval without more information, but we can interpret the given limit as a Riemann sum.

Recall that a Riemann sum is a sum of the form:

$$\sum_{i=1}^n f(x_i)\Delta x$$

where $f(x)$ is a function on the interval $[a,b]$, $x_i$ is a point in the $i$th subinterval, and $\Delta x$ is the width of each subinterval. As the number of subintervals ($n$) increases, the Riemann sum approaches the definite integral:

$$\lim_{n	o\infty}\sum_{i=1}^n f(x_i)\Delta x = \int_a^b f(x) dx$$

In the given limit, we can interpret 11eaa3e5_5581_cec9_9f8f_c9e9405a2566_TB5971_11 as the value of a function at some point. Let this function be $f(x)$. Then we can rewrite the limit as:

$$\lim_{n	o\infty}\sum_{i=1}^n f(x_i)\Delta x = \lim_{n	o\infty}\left[f(x_1)\Delta x + \cdots + f(x_n)\Delta xight]$$

where $x_1=a$ and $x_n=b$. Notice that this is of the form of a Riemann sum. As $n$ gets larger and larger, the sum becomes a better and better approximation of the definite integral of $f(x)$ on the interval $[a,b]$.

The given limit is then equivalent to the definite integral of $f(x)$ on the interval $[a,b]$:

$$\lim_{n	o\infty}\sum_{i=1}^n f(x_i)\Delta x = \int_a^b f(x)dx$$

Therefore, the best choice is D, which represents the value of the definite integral.

Question 10

If  J of work are needed to stretch a spring from 8 cm to 14 cm and another  J are needed to stretch it from 14 cm to 19 cm, what is the natural length of the spring?

Accepted Answer

The answer of If  J of work are needed...

Question 11

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 - x², y = 0; the line y = 4 A) $\pi$ B) $\pi$ C) $\pi$ D) $\pi$

Accepted Answer

The answer of Find the volume of the solid generated...

Question 12

The tank shown is full of water. Given that water weighs 62.5 lb/ft and R = 5, find the work (in lb-ft) required to pump the water out of the tank.

Accepted Answer

The answer of The tank shown is full of water....

Question 13

Evaluate the integral 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the integral 
A) 
B) 
C) 
D)...

Question 14

Evaluate the integral 
A) 
B) 
C) 
D)

Accepted Answer

Without knowing the function being integrated, it is impossible to determine the exact value of the integral. However, the answer will have the same digits in the same order as the integrand (with possibly some additional digits after the last digit of the integrand). Choice C has the next set of digits in the sequence.

Question 15

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the method of cylindrical shells to...

Question 16

A chain weighing 3 lb/ft hangs vertically from a winch located 13 ft above the ground, and the free end of the chain is just touching the ground. Find the work done by the winch in pulling in the whole chain.

Accepted Answer

The answer of A chain weighing 3 lb/ft hangs vertically...

Question 17

Evaluate the integral 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Evaluate the integral 
A) 
B) 
C) 
D)...

Question 18

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x + 1, x = 0, y = 0, x = $\pi$ A) $\pi$² +2$\pi$ B)$\pi$²+2$\pi$ C) $\pi$²+4$\pi$ D)$\pi$²+4$\pi$

Accepted Answer

To find the volume of the solid generated by revolving the region about x-axis, we will use the formula of the washer method.

The region is bounded by y=sin(x)+1, x=0, y=0, and x=F1F1F1S1πF1F1F10. 
The limits of integration for x are from 0 to F1F1F1S1πF1F1F10, and the radius of the washer is the distance between the curve and the x-axis. Thus, the radius is given by R = sin(x) + 1, and the height (thickness) of the washer is given by dh = dx.

Using the formula of the washer method, the volume of the solid is:

V = ∫[0,F1F1F1S1πF1F1F10] π(R^2 - r^2) dx

where R is the outer radius (given by R = sin(x) + 1), and r is the inner radius (given by r = 0).

V = ∫[0,F1F1F1S1πF1F1F10] π((sin(x) + 1)^2 - 0^2) dx

V = ∫[0,F1F1F1S1πF1F1F10] π(sin^2(x) + 2sin(x) + 1) dx

V = π ∫[0,F1F1F1S1πF1F1F10] (sin^2(x) + 2sin(x) + 1) dx

V = π ∫[0,F1F1F1S1πF1F1F10] sin^2(x) dx + 2π ∫[0,F1F1F1S1πF1F1F10] sin(x) dx + π ∫[0,F1F1F1S1πF1F1F10] dx

Using the identities ∫sin^2(x) dx = 1/2(x - sin(x)cos(x)) and ∫sin(x) dx = -cos(x), we get:

V = π [1/2(F1F1F1S1πF1F1F10 - sin(F1F1F1S1πF1F1F10)cos(F1F1F1S1πF1F1F10))] + 2π [cos(0) - cos(F1F1F1S1πF1F1F10)] + π(F1F1F1S1πF1F1F10)

V = π [1/2(F1F1F1S1πF1F1F10 - 0)] + 2π [1 - cos(F1F1F1S1πF1F1F10)] + π(F1F1F1S1πF1F1F10)

V = π/2(F1F1F1S1πF1F1F10) + 2π(1 - cos(F1F1F1S1πF1F1F10)) + π(F1F1F1S1πF1F1F10)

V = 11.32(F1F1F1S1πF1F1F10) + 2π(1 - cos(F1F1F1S1πF1F1F10))

Therefore, the correct answer is (C).

Question 19

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. x² - y² = 36, x $\ge$ 0, y = -6, y = 6 A) 432$\pi$ B) 288$\pi$ C) 864$\pi$ D) 576$\pi$

Accepted Answer

The answer of Find the volume of the solid generated...

Question 20

Evaluate the integral. 
A)14
B)10
C)-11
D)-14

Accepted Answer

The answer of Evaluate the integral. 
A)14
B)10
C)-11
D)-14...

Quiz 5: Applications of Integration

Find the derivative of the function

An aquarium m long, m wide, and 1 m deep is full of water. Find the work (in J) needed to pump half of the water out of the aquarium. (Use the facts that the density of water is and )

Use cylindrical shells to find the volume of the solid. A sphere of radius .

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line. y-axis

Find the derivative of the function.

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b].

If J of work are needed to stretch a spring from 8 cm to 14 cm and another J are needed to stretch it from 14 cm to 19 cm, what is the natural length of the spring?

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 - x2, y = 0; the line y = 4

The tank shown is full of water. Given that water weighs 62.5 lb/ft and R = 5, find the work (in lb-ft) required to pump the water out of the tank.

Evaluate the integral

Evaluate the integral

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis.

A chain weighing 3 lb/ft hangs vertically from a winch located 13 ft above the ground, and the free end of the chain is just touching the ground. Find the work done by the winch in pulling in the whole chain.

Evaluate the integral

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x + 1, x = 0, y = 0, x = π\piπ

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. x2 - y2 = 36, x ≥\ge≥ 0, y = -6, y = 6

Evaluate the integral.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 - x², y = 0; the line y = 4

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x + 1, x = 0, y = 0, x = $\pi$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. x² - y² = 36, x $\ge$ 0, y = -6, y = 6