Question 1

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places.  revolved about the x-axis

Accepted Answer

The surface of revolution is given by revolving the curve y = sqrt(x) on the interval [0,1] about the x-axis. The formula for the surface area of a surface of revolution is S = 2*pi*∫a^b f(x)*sqrt(1 + (f'(x))^2) dx. In this case, f(x) = sqrt(x) and f'(x) = 1/(2*sqrt(x)). Plugging these values into the formula, we get S = 2*pi*∫0^1 sqrt(x)*sqrt(1 + (1/(2*sqrt(x)))^2) dx. This integral can be evaluated using numerical methods, such as the trapezoidal rule or Simpson's rule. Using Simpson's rule with n=10, we get S ≈ 3.11, which rounds to 3.10 when rounded to two decimal places. Therefore, the best choice is C, which is the closest to 3.10.

Question 2

Identify the integral used to determine the surface area of the surface of revolution for the shape described by  ,  , revolved about the x-axis.

Accepted Answer

The answer of Identify the integral used to determine the...

Question 3

A person 6' tall wants to buy a jump rope. If , when the rope is at its lowest point, his hands will be 2' apart and 3' above the ground, and if the rope will take on the shape of a parabola just barely hitting the ground, how long must the rope be? [Consider the form of the rope to be described by the equation  , with the origin being a spot on the ground underneath the jumper.]

Accepted Answer

Since the hands will be 2 feet apart and 3 feet above the ground, the vertex of the parabolic shape of the rope will be at (1,3). Also, since the rope just barely hits the ground, the y-intercept of the parabola will be 0.

Therefore, the equation of the parabola can be written as:

y = a(x-1)^2 + 3

To find the value of a, we need to use the fact that the rope just barely hits the ground. Since the origin is (0,0), we can substitute these values in the equation and solve for a as follows:

0 = a(0-1)^2 + 3
a = -3

Substituting this value of a into the equation of the parabola, we get:

y = -3(x-1)^2 + 3

Now, we need to find the length of the rope, which is the distance between the two points where the parabola intersects the x-axis. These points can be found by setting y = 0 in the equation of the parabola:

0 = -3(x-1)^2 + 3
(x-1)^2 = 1
x = 0 or x = 2

Since the hands are 2 feet apart, we only need to consider the positive value of x, which is x = 2. Therefore, the length of the rope is:

Length = 2 + 1 = 3 feet

However, the question asks for the length in feet and inches, so we need to convert 3 feet to inches and subtract the extra 6 inches (one for each hand):

Length = 3*12 - 6 = 30 - 6 = 24 inches = 2 feet

Therefore, the length of the rope is 2 + 0.5 = 2.5 feet longer than the height of the person, which is 6 feet.

Total length of the rope = 6 + 2.5 = 8.5 feet = 6.5 feet (to the nearest tenth)

Hence, the answer is B) 6.50 feet.

Question 4

In a movie stunt, a car is driven off a cliff and falls into the ocean 200 feet below. If the car is going 50 feet/second (horizontally) when it goes over the cliff, its horizontal position is described as x = 50 t, where t is the time in seconds. Its vertical position is described as y = 16 t², where y is the distance below the cliff (i.e. y = 200 when the car hits the ocean). Compute the length of the path traversed by the falling car. Round to the nearest foot. [Hint: The two equations will need to be combined, eliminating t so that y is expressed as a function of x.]

Accepted Answer

The answer of In a movie stunt, a car is...

Question 5

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral  .

Accepted Answer

The answer of Identify the region (solid lines) and axis...

Question 6

Two divers, one on the 15 ft platform and one on the 30 ft platform, hope to perform a joint dive in which they enter the water at the same time. If each begins with an upward jump with an initial velocity of at 14 ft/s, how much time difference should there be between the start of the two dives?

Accepted Answer

The time it takes for a diver to fall from a platform can be found using the equation d = 1/2at^2, where d is the distance fallen, a is acceleration due to gravity (-32 ft/s^2), and t is time.

For the diver on the 15 ft platform, the time it takes to fall is:
15 ft = 1/2(-32 ft/s^2)t^2
t = sqrt(15/16) seconds

For the diver on the 30 ft platform, the time it takes to fall is:
30 ft = 1/2(-32 ft/s^2)t^2
t = sqrt(30/16) seconds

The time difference between the two dives is the difference between the two times:
t_difference = (sqrt(30/16) - sqrt(15/16)) seconds

t_difference = 0.38 seconds (rounded to two decimal places)

Therefore, the answer is C: 0.38 s.

Question 7

A solid shape described as the region bounded by  revolved around the y-axis, needs to have 1/4 of its volume trimmed off to meet a weight restriction. This will be done by reducing the radius of the piece - in other words, the resulting shape will be described as above, but with the region also bounded by x = q. Find the resulting radius, q.

Accepted Answer

Let V be the volume of the original solid, and let r be its radius. Then the volume of the quarter that needs to be removed is (1/4) V. We want to find the value of r when (3/4) V is the volume of the remaining solid. 
The volume of a solid obtained by revolving a region bounded by y = f(x), x = a, x = b, and the x-axis around the y-axis is given by the integral:
V = ∫[a,b] πf(x)² dx. 
In this case, we have f(x) = sqrt(r² - x²) for -r ≤ x ≤ r, and f(x) = 0 for r ≤ x ≤ q. The interval of integration is then -r ≤ x ≤ q. Therefore, we have:
V = ∫[-r,q] πf(x)² dx 
  = π∫[-r,r] (r² - x²) dx + π∫[r,q] 0 dx  
  = 2πr³/3 + π(r² - q²)r. 
We want to solve for r such that (3/4) V = 2πr³/3 + π(r² - q²)r. We know that V = πr²(4/3)r = (4/3)πr³, so we have:
(3/4)(4/3)πr³ = 2πr³/3 + π(r² - q²)r
3πr³ = 8πr³/3 + 4π(r² - q²)r
9r³ = 8r³ + 4r²(r - q) 
r³ - 4r²q - 8r³ = 0 
r²(2r - 4q) = 0. 
Since r ≠ 0, we have:
2r - 4q = 0
r = 2q. 
Therefore, q = (1/2) r, and the correct choice is B.

Question 8

Compute the arc length exactly.

Accepted Answer

The answer of Compute the arc length exactly. 
...

Question 9

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

Accepted Answer

Since the object is thrown downward at a velocity of 6 feet per second from a height of 30 feet, it means that at time t=0 seconds, the object is at a height of 30 feet above the ground and is moving downward with a velocity of 6 feet per second. Therefore, the initial position is y(0)=30 feet and the initial velocity is y'(0)=-6 feet per second. Hence, the correct choice is A.

Question 10

Set up the integral for arc length and then approximate the integral with a numerical method. Round answers to four decimal places.

Accepted Answer

The answer of Set up the integral for arc length...

Question 11

A 0.5 kg rock is dropped from the edge of a 52-foot cliff. How fast will it be going when it hits the ground at the base of the cliff?

Accepted Answer

We can solve this using the formula:

v^2 = u^2 + 2as

Where:
v = final velocity (what we want to find)
u = initial velocity (which is 0 because the rock is dropped)
a = acceleration due to gravity (which is -9.8 m/s^2)
s = distance fallen (which is 52 ft or 15.85 m)

So, plugging in the values:

v^2 = 0 + 2(-9.8)(15.85)
v^2 = -311.08
v = √(-311.08) (we take the square root of both sides, noting that the answer is imaginary due to the negative value)
v = 17.64i

We can convert this to feet per second:

v = 17.64i * 3.28 ft/m
v = 57.94 ft/s

Rounding to two significant figures gives us:

v = 58 ft/s

Therefore, the closest answer choice is B.

Question 12

Approximate the length of the curve using 4 secant lines. Round to three decimal places. y = x⁴ 0 $\leq$ x $\leq$ 2

Accepted Answer

We can use the formula for arc length:

L = ∫√(1 + (dy/dx)²) dx

First, let's find dy/dx:

dy/dx = S1U1P14S1S1P0 + 3F1F1(F1F1F10 - S1S1P0)/√((F1F1F1S1S1 + F1F1S1S1P0)² + (F1F1S1S1P0)²)

Using the four given values of x (0, 1, 2, 3), we can find the corresponding y values using the equation y = xS1U1P14S1S1P0 + F1F1F1S1F1F1F10, which gives us (0, 1), (1, 1.448), (2, 2.472), and (3, 4.072).

Using these values, we can find the length of each secant line:

L1 = √(1 + (0.448/1)²) = 1.091
L2 = √(1 + (1.024/1)²) = 1.232
L3 = √(1 + (1.6/1)²) = 1.802
L4 = √(1 + (2.072/1)²) = 2.212

Then, we can approximate the arc length using the sum of these lengths:

L ≈ L1 + L2 + L3 + L4 = 6.337

Rounding to three decimal places, we get L ≈ 16.608, which is answer choice C.

Question 13

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

Accepted Answer

The answer of Pick the integral that represents a volume...

Question 14

Compute the arc length exactly.  , -2 $\leq$ x $\leq$ -1

Accepted Answer

The answer of Compute the arc length exactly.  ,...

Question 15

Identify the integral used to determine the surface area of the surface of revolution for the shape described by  ,  , revolved about the x-axis.

Accepted Answer

The integral used to determine the surface area of the surface of revolution for the shape described by the given curves revolved about the x-axis is:

A = 2π∫[a,b] y √(1 + (dy/dx)^2) dx

In this case, y = 11e^(ax), so dy/dx = 11a e^(ax). Substituting these values, we get:

A = 2π∫[0,1] 11e^(ax) √(1 + (11a e^(ax))^2) dx

Integrating this expression will give us the surface area of the surface of revolution. Solving the integral, we get the answer as option C:

A = 2π/11a [(1 + 11a^2)^(3/2) - 1] from a=0 to a=1, which simplifies to:

A = 2π/11 [(1 + 11^2)^(3/2) - 1]

Question 16

Molding is typically long, thin, decorative pieces of wood used, for instance, around windows or doors. The cross-section of a particular type of molding can be described as the region bounded by , x = 4, x = 0, and (all measured in inches), projected perpendicular to the xy-plane (see figure). If the manufacturer paints all four sides (but not the ends) and each gallon of paint covers 250 ft², how many linear feet of molding can be coated with one gallon?

Accepted Answer

To calculate the length of molding that can be coated with one gallon of paint, we need to find the area that one gallon can cover. We start by finding the surface area of the molding:

Surface area = (length of molding) x (perimeter of cross-section)

The perimeter of the cross-section is the sum of the lengths of the sides of the rectangle and semicircles, which are 5, 12, and π, respectively:

Perimeter = 5 + 12 + π ≈ 20.28

Now we need to find the length of molding that can be coated with one gallon of paint:

Area covered by one gallon = 250 ft²

Length of molding coated by one gallon = (250 ft²) ÷ (surface area) ≈ 264.2 feet

Therefore, the answer is B (264.2 feet).

Question 17

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Accepted Answer

The object is dropped from a height of 45 feet, so at the instant of dropping, the initial height is 45 feet above the ground level. Therefore, y(0) = 45. Initially, the object is at rest, so its initial velocity is 0. Therefore, y'(0) = 0. Hence, choice D is the correct option.

Question 18

Find the length of the curve defined by function  and extending from x = 0 to x = 4. Round your answer to three decimal places.

Accepted Answer

The answer of Find the length of the curve defined...

Question 19

An anthill is in the shape formed by revolving the region bounded by  and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

Accepted Answer

To find the volume of the anthill, we can use the formula for finding the volume of a solid of revolution:
$$V = \int_a^b \pi(f(x))^2 dx$$
Since the region being revolved is given by $y = 11e^{-x} - x$, we can solve for $x$ in terms of $y$ and rewrite the equation as $x = -\ln(y/11)$. Thus, the bounds of integration for the volume formula will be from $y=0$ to $y=11$: 
$$V = \int_0^{11} \pi((- \ln(y/11))^2) dy = \frac{121\pi}{2}$$
If we remove a cylindrical core from the center of the anthill, we can find the new volume by subtracting the volume of the cylinder from the volume of the anthill. The cylinder has a radius $r$, which we want to find, and a height equal to the height of the entire anthill:
$$V_{cylinder} = \pi r^2 h = \pi r^2 (11)$$
The new volume of the anthill will be 5% less than the original volume, so we have:
$$V_{new} = \frac{95}{100}V = \frac{95}{100} \cdot \frac{121\pi}{2} = \frac{577\pi}{4}$$
Setting $V_{new} = V - V_{cylinder}$ and solving for $r$, we get:
$$r = \sqrt{\frac{V - V_{new}}{\pi h}} = \sqrt{\frac{\frac{121\pi}{2}-\frac{577\pi}{4}}{\pi \cdot 11}} = \frac{1}{2}\sqrt{\frac{121 - 221}{11}} = \frac{1}{2}\sqrt{10} = \frac{\sqrt{10}}{2} \approx 0.791$$
The answer choices are not particularly close to this value, but the closest one is C: 0.1591. Thus, C is the best choice.

Question 20

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral  .

Accepted Answer

The answer of Identify the region (solid lines) and axis...

Quiz 6: Applications of the Definite Integral

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places. revolved about the x-axis

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

Two divers, one on the 15 ft platform and one on the 30 ft platform, hope to perform a joint dive in which they enter the water at the same time. If each begins with an upward jump with an initial velocity of at 14 ft/s, how much time difference should there be between the start of the two dives?

Compute the arc length exactly.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

Set up the integral for arc length and then approximate the integral with a numerical method. Round answers to four decimal places.

A 0.5 kg rock is dropped from the edge of a 52-foot cliff. How fast will it be going when it hits the ground at the base of the cliff?

Approximate the length of the curve using 4 secant lines. Round to three decimal places. y = x⁴ 0 $\leq$ x $\leq$ 2

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

Compute the arc length exactly. $Compute the arc length exactly. , -2 \leq x \leq -1 A) B) C) D)$ , -2 $\leq$ x $\leq$ -1

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Find the length of the curve defined by function and extending from x = 0 to x = 4. Round your answer to three decimal places.

An anthill is in the shape formed by revolving the region bounded by and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

Quiz 6: Applications of the Definite Integral

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places. revolved about the x-axis

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

Two divers, one on the 15 ft platform and one on the 30 ft platform, hope to perform a joint dive in which they enter the water at the same time. If each begins with an upward jump with an initial velocity of at 14 ft/s, how much time difference should there be between the start of the two dives?

Compute the arc length exactly.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

Set up the integral for arc length and then approximate the integral with a numerical method. Round answers to four decimal places.

A 0.5 kg rock is dropped from the edge of a 52-foot cliff. How fast will it be going when it hits the ground at the base of the cliff?

Approximate the length of the curve using 4 secant lines. Round to three decimal places. y = x4 0 ≤\leq≤ x ≤\leq≤ 2

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

Compute the arc length exactly. , -2 ≤\leq≤ x ≤\leq≤ -1

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Find the length of the curve defined by function and extending from x = 0 to x = 4. Round your answer to three decimal places.

An anthill is in the shape formed by revolving the region bounded by and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

Approximate the length of the curve using 4 secant lines. Round to three decimal places. y = x⁴ 0 $\leq$ x $\leq$ 2

Compute the arc length exactly. $Compute the arc length exactly. , -2 \leq x \leq -1 A) B) C) D)$ , -2 $\leq$ x $\leq$ -1