The integral as a limit of Riemann sums using right endpoints is =lim (n→∞) n ∑i=1[(ln(1+(4+8i/n)2)−sin(4+8i/n))8/n].
Definite integralThe definite integral is defined here in terms of its limit.
So the definite integral goes as follows.
∫f(x)dx= lim(n→∞) n∑i=1 f(xi)Δx.
Where, for each positive integer n, we let Δx=(b−a) /n
And in case of i=1,2, 3..., n, we let xi=a+iΔx.
For Δx
For each n, we get
Δx=(b−a)/n= (12−4)/n=8/n
For xi
Xi = a+iΔx = 4+i8/n=4+8i/n
For f(xi)
∫124 [ln(1+x2) −sin x] dx
=lim (n→∞) n ∑i=1[(ln(1+(4+8i/n)2)−sin(4+8i/n))8/n]
What is integral?The Riemann integral, developed by Bernhard Riemann, was the first exact formulation of the integral of a function over an interval in the field of mathematics known as real analysis.
It was presented to the faculty at the University of Göttingen in 1854, but it wasn’t until 1868 that it was published in a journal.
The Riemann integral can be found using the calculus fundamental theorem for a variety of functions and real-world situations, or it can be roughly approximated using numerical integration.
The Riemann integralThe Riemann integral can be thought of as the limit of a function’s Riemann sums as the finer the partitions are.
The function is referred to as being integrable if the limit exists (or more specifically Riemann-integrable). By making the partition fine enough, the Riemann sum can be brought as close as desired to the Riemann integral.
The mesh of the partitions must, among other things, get smaller and smaller until, at the limit, it is zero.
If this were not the case, then on specific subintervals, we would not be receiving a fair approximation to the function. Actually, this is sufficient to define an integral.
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Tom has a meter stick which is marked off in centimeters (cm). he measures a distance three (3)
times and gets measurements of: 11.52m, 11.51m, and 11.51m. a later calibration of the stick
shows it to be actually 1.001m long (1mm too long). tom's friend bill also has a meter
stick but
it's marked off in millimeters (mm). he measures the same distance three times also, and gets:
11.452m, 11.451m, and 11.451m. a later calibration of his stick shows it to be 1.05m long (5 cm
too long)
a.
whose measurements are more precise, tom's or bill's?
b. whose measurements are more accurate?
a. The measurement done by Bill is more precise than Tom. Because all the measurements done by Bill are measured up to mm and which are very close to each other, but for Tom's measurement value measured up to cm.
b. The measurement done by Tom is more accurate than Bill's measurement. Because the actual scale of meter scale is close to 1.001m, which is off by 1mm, but Bill's meter stick is 1.05m, which have errors in centimeters.
What is meter scale?Length is one of the fundamental units in system of measurement. The SI unit of the length is meter. The meter scale is apparatus for measurement of the length of any object. Example: The length of square is 5 meters. Here, 5 is magnitude of the length and meter is the unit of length. 1 meter of meter scale is divided into 100 centimeters. Meter scale is used to measure the Length.
Note: The meter scale reading does not change with the temperature and other physical quantities. The scale used for measurement should not be affected by the physical environment, it should be available easily. Thus, to make any device for measurement of physical quantity above criteria should be kept in mind.
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2. what is the unit of force in metric system? a. calorie b. joules c. newtons d. pounds
The unit of force in metric system is Newtons
A metric system is used for measuring things using units like mm, cm, m and km for length, g and kg for weight, s for time etc.
Newton can be defined as 1 kg m / [tex]s^{2}[/tex].
where,
kg = Kilogram
m = Meter
s = second
As we can clearly see all these units are part of metric system.
Therefore, the unit of force in metric system is Newtons
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A bike accelerates from rest at 4 m/s/s. what is the velocity 3.5 s after the bike starts moving?
A motorbike starts from rest and speeds up uniformly to the right until it reaches a maximum velocity of 60 m/s in 15 s. what is the acceleration of the motor bike?
The acceleration of the motor bike is 4 m/s².
To find the acceleration, the given values are,
Final velocity = 60 m/s
Initial velocity = 0 m/s
Time period = 15 seconds.
What is Acceleration?Acceleration is a rate of change of velocity with respect to time with respect to direction and speed.A point or an object moving in a straight line is accelerated if it speeds up or slows down.Acceleration formula can be written as,a = (vf - vi) / t m/s²
Acceleration = (change in speed) / (time for the change)
a = acceleration
vf = final velocity
vi= starting velocity
t = time
Acceleration is a vector quantity.The unit of acceleration is m/s².So the formula,
a = Δv / t m/s²
= (60 m/s) / (15)
= 4 m/s².
The acceleration of the motor bike is 4 m/s².
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with what speed do the molecules move towards the piston if it moves down?
The volume of the space would decrease if the piston were to quickly move out, and the process would then reverse itself, taking another hundredth of a second to complete before everything was back to normal. The time it takes for the molecules to balance out is constant, regardless of how quickly or how far the piston is driven.
In mechanical engineering, a piston and cylinder are two sliding cylinders with closed heads that move reciprocally within a somewhat larger cylindrical chamber under pressure from a fluid, as in an engine or pump. The piston rod, which is rigidly coupled to the piston, can pass through one of the end cover plates of a steam engine's cylinder, which is closed by plates at both ends, using a gland and stuffing box (steam-tight joint).
An internal combustion engine's cylinder is open at one end to allow the connecting rod, which connects the piston to the crankshaft, to freely oscillate and is closed at one end by a plate known as the head.
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How far does a freely falling apple drop in 7.4 s after being released from rest?
The freely falling apple will travel 268.32 meters along y-axis
We need to know about free fall motion to solve this problem. The free fall motion should follow this rule
vt = gt
vt² = 2gh
h = 1/2 . g . t²
where vt is final velocity, g is the gravitational acceleration (9.8 m/s²), h os height and t is time.
From the question above, we know that
t = 7.4 s
Use the third equation to get the distance (height)
h = 1/2 . g . t²
h = 1/2 . 9.8 . 7.4²
h = 268.32 m
The freely falling apple will travel 268.32 meters along y-axis.
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A fan blows the air and gives it kinetic energy. an hour after the fan has been turned off, what has happened to the kinetic energy of the air?
A fan blows the air and gives it kinetic energy, an hour after the fan has been turned off, the kinetic energy turns into thermal energy.
The fact that all energy is conserved causes a heat flow, which raises internal (or thermal) energy, as the air molecules slow and lose kinetic energy.
Kinetic energy, which may be seen in the movement of an object, particle, or group of particles, is the energy of motion. Any moving item uses kinetic energy, such as a person walking, a baseball being thrown, a piece of food falling from a table, or a charged particle in an electric field.
The energy of a body in motion, or kinetic energy (KE), is essentially the energy of all moving objects. Along with potential energy, which is the stored energy present in objects at rest, it is one of the two primary types of energy.
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The mass of an apple is 84g and the volume is 120 cm^3. Calculate the density in g/cm^3
Answer:
0.7 g/cm^3
Explanation:
Density = Mass / Volume
= 84g / 120 cm^3
= 0.7 g / cm^3
Compare the melting point and the melting temperature range of a pure substance and an impure substance. for a pure substance, the melting temperature range is:________
For a pure substance, the melting temperature range is low
The melting point range is widened as a result of impurity addition, which lowers the melting point. The melting point range is reduced because pure substances have greater melting points.
The typical definition of the melting point is the temperature at which a substance transforms from a solid to a liquid.
The melting point of a liquid is the temperature at which the liquid transforms from a solid to a liquid under atmospheric pressure. This is the location where the liquid and solid phases are equally present.
By placing a tiny amount of an organic solid into a tiny capillary tube, affixing it to the stem of a thermometer centred in a heating bath, slowly heating the bath, and observing the temperatures at which melting begins and is complete, it is possible to determine the melting point of an organic solid.
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An object weighing 1.840 kg has a volume of 0.0015 m 3. what is the density of the object in g/cm3?
Answer:
1,226,666.6666 g/cm3
Explanation:
Density =Mass/Volume
=1.840 kg(×1000g) / 0.0015
=1,226,666.6666666 g /cm3
Three liquids have densities of 1.15 g/ml, 0.79 g/ml and 0.95 g/ml. when the liquids, which do not mix, are poured into a graduated cylinder, the liquid at the top is the one with a density of:________
The liquid at the top is the one with a density of 0.79 g/ml.
The matter with most density will stay at the bottom of the graduated cylinder. So when comparing the three given densities of 1.15 g/ml, 0.79 g/ml and 0.95 g/ml, the lowest density is 0.79 g/ml.
Density is nothing but weight of the matter. So basically the lighter one stays at top.
Therefore, the liquid at the top is the one with a density of 0.79 g/ml.
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A cyclist rides at an average speed of 23 miles per hour. if she wants to bike 190 km , how long (in hours) must she ride?
An average cyclist travels at a speed of 23 mph. Time (t) = 5.1 hours if she wants to ride 190 kilometers.
What is speed?The distance traveled in a unit of time is called speed. It refers to a thing's rate of movement. The scalar quantity known as speed is the velocity vector's magnitude. It has no clear direction. A faster rate of movement is indicated by a higher speed. It travels slower if the speed is lower. It has zero speed if it isn't moving at all.
The formula: can be used to determine the constant velocity of an object traveling straight forward.
r = d / t
where
r stands for rate or speed (sometimes denoted as v, for velocity)
d is the traveled distance.
t the amount of time needed to finish the movement
The average speed of an object over a period of time is given by this equation. We can see the object's average speed here, however at some moments over the time frame, it may have been moving faster or slower.
As the time interval gets closer to zero, the maximum average speed is reached. The speed at that moment is what you see when you look at a speedometer in an automobile. While you may have briefly reached a speed of 60 mph, your average speed over the course of 10 minutes could have been much higher or much lower.
190 Km (given)
1 km = 0.621 miles (should know)
So, 190 km = (190x0-621) miles
= 118 miles
23 miles per hour means,
23 miles- 1 hour
so, 118 miles - (118 × 1/ 23) hour
118 miles = 5.1 hour.
Time (t) = 5.1 hour
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An aluminum rod of e = 70 gpa and a diameter of 15mm exhibited a strain value of 0.001 mm/mm when subjected to a load p. the value of the load p is most nearly:_______
a. 15000
b. 12000
c. 10700
d. 12370
By elasticity, the most nearly load is D. 12370 N .
We need to know about elasticity to solve this problem. In elasticity, stress can be determined as
σ = e . δ
where σ is stress, e is elastic modulus and δ is the strain
From the question above, we know that:
e = 70 GPa = 70 x 10⁹ Pa
d = 15 mm = 0.015 m
δ = 0.001 mm/mm
Find the stress
σ = e . δ
σ = 70 x 10⁹ . 0.001
σ = 7 x 10⁷ Pa
Find the surface area of the aluminum rod
A = πR²
A = π(0.015/2)²
A = 1.77 x 10¯⁴ m²
Find the load
F / A = σ
F = σ x A
F = 7 x 10⁷ x 1.77 x 10¯⁴
F = 12375 N
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Calculate the wavelengths in air at 20∘c for sounds in the maximum range of human hearing. The speed of sound in air is 343 m/s.
The wavelengths in air is 8.8x10^-6 m for sounds in the maximum range of human hearing.
Wavelength is the distance between identical points (adjacent crests) in the adjacent cycles of a waveform signal propagated in space or along a wire.
The following equation can be used to compute wavelength: wavelength = wave velocity/frequency. In most cases, wavelength is measured in metres. Greek letter lambda is used to represent wavelength, so = v/f.
speed = wavelength x frequency or wavelength = speed / freq
for f = 20Hz, wavelength = 343m/s / 2.0 x 10^1 Hz = 17.2m
for f = 20,000 Hz, wavelength = 343m/s / 20,000Hz = 0.017 m
39MHz = 39x10^6 Hz so wavelength = 343m/s / 39x10^Hz = 8.8x10^-6 m
Hence, the wavelengths in air is 8.8x10^-6 m
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A 120.0 l weather balloon filled with 4.90 mol of helium has a small leak. if the helium leaks at a rate of 10.0 mmol/hr, what is the volume of the balloon after 67.0 hours?
The volume of the balloon after 67 hours would be 119.98 L
Calculationmol of he = 4.90 mol
wt. of he = 4.90x 4 = 19.60 g
after 67 hours,
67 x 10 x [tex]10^{-3}[/tex] x 4 = 2.68 g
we know 1 L = 1000 g
so, volume of gas = 2.68 x [tex]10^{-3}[/tex] L
amount left = 16.92 g = 16.92 x [tex]10^{-3}[/tex] L
thus, volume after 67 hours = 119.98 L
What is a weather balloon?A weather balloon, or also known as the sounding balloon, is a specially crafted balloon that is used to carry instruments above to the sky to send back information to the radio signal centres about atmospheric pressure, temperature, humidity and wind speed by means of a small, expendable measuring device called a radiosonde.
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What is the correct number of p orbital electrons in cyclopentadienyl anion?
a) 6
b) 2
c) 5
d) 4
The number of p - orbital electrons in cyclopentadienyl anion are
6 electrons.
We have Cyclopentadienyl anion.
We have to determine number of p - orbital electrons in it.
What are Orbitals ?Atomic orbitals are the 3-dimensional spaces around the nucleus where the probability of finding an electron is maximum.
According to the question -
The Cyclopentadienyl Anion is planar and has a continuous overlap of 5 p-orbitals with 6 electrons. It is an aromatic system which is highly stable and has a closed shell configuration. The deprotonated form of cyclopentadiene is represented as the cyclopentadienyl anion C5H5⁻.
Hence, the number of p - orbital electrons in cyclopentadienyl anion are
6 electrons.
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The vernal equinox is now just entering the constellation aquarius. In what constellation will it lie in the year a. D. 10000?.
The vernal equinox in the year AD 10000 will lie in the constellation Scorpio.
Once every 25800 years a full cycle of precession is completed. There are 12 constellations which means it takes 25800 / 12 = 2150 years to move from one constellation to the next one. The order in which the vernal equinox moves from Aquarius is Capricorn, Sagittarius, Scorpio, Libra etc.
Now the year is 2022 AD where the vernal equinox just entered constellation Aquarius. In 4172 AD it would have entered constellation Capricorn, then constellation Sagittarius in 6322 AD, then constellation Scorpio in 8472 AD and then constellation Libra in 10622 AD. This means that in 10000 AD it is still in constellation Scorpio and yet to enter constellation Libra.
Therefore, In the year 10000 AD, the vernal equinox will lie in the constellation Scorpio.
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A car can go from 0 to 60 mph in 6.0 s .assuming that it could maintain the same acceleration at higher speeds, how long would it take the car to go from 0 to 120 mph?
The time taken for the car to reach 120 mph speed is 12 seconds.
Acceleration of the carThe acceleration of the car is the rate of change of velocity of the car with time.
a = Δv/t
where;
t is time = 6 sa = (60 mph) / (6 s)
when the speed increases to 120 mphThe time of motion at the given constant speed is calculated as follows;
t = v/a
t = (120 mph) / (60 mph / 6s)
t = (120 mph x 6 s) / (60 mph)
t = 2 x 6 s
t = 12 seconds
Thus, the time taken for the car to reach 120 mph speed is 12 seconds.
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a skateborder traviling at 7.0m/s/s rolls to a stop at the top of a ramp in 3.0 seconds what is the skateborder's acceleration?
Answer:
-2.33 m/s^2
Explanation:
initial speed u = 7m/s
final speed v= 3s
acceleration a = (v-u)/t
a= -7/3= -2.33 meter per second square.
If a sprinter reaches his top speed of 10.9 m/s in 2.84 s , what will be his total time?
If a sprinter reaches his top speed of 10.9 m/s in 2.84 s , his total time to cover a distance of 100 m is 10.59 sec
We know that,
a = u / t
s = ut + 1/2 a[tex]t^{2}[/tex]
where,
s = Distance
u = Initial velocity
a = Acceleration
t = Time
Given that,
Top speed = 10.9 m / s
Time taken to reach top speed = 2.84 s
a = 10.9 / 2.84 = 3.83 m / [tex]s^{2}[/tex]
During the time period of 2.84 s, the distance covered is ,
s = ( 0 * 2.84 ) + 1/2 ( 3.83 * [tex]2.84^{2}[/tex] ) = (3.83 * 8.07) / 2
s = 15.44 m
Consider the sprinter maintains a constant acceleration after reaching his top speed.
The distance remaining in 100 m track = 100 - 15.44 = 84.56 m
Time taken to cover 84.56 m at 10.9 m / s with no acceleration is
84.56 = ( 10.9 * t ) + 1 / 2 ( 0 ) ( [tex]t^{2}[/tex] )
t = 7.75 s
Total time taken = 7.75 + 2.84 = 10.59 s
Acceleration is the rate of change of velocity. It is represented as a. Its unit is m / [tex]s^{2}[/tex]
Therefore, if a sprinter reaches his top speed of 10.9 m/s in 2.84 s , his total time to cover a distance of 100 m is 10.59 sec
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If an electron begins from rest and is accelerated by a field of 0.060 n/c, what is its speed when it collides with a metal atom?
The speed when it collides with a metal atom = 27.5m/s
What is kinetic energy?A particle or item in motion has a form of energy called kinetic energy. When work, which entails the energy transfer is done on an object by applying a net force, that object acquires kinetic energy.
In the electric field, the electron loses potential energy while gaining kinetic energy. You may calculate the kinetic energy as
[tex]K=\frac{m_e v^2}{2}=e E d[/tex]
Here,
[tex]\text { } m_e=9.11 \cdot 10^{-31} \mathrm{~kg}[/tex] is the mass of the electron;
[tex]e=1.6 \cdot 10^{-19} C[/tex] is the charge of the electron;
E is the electric field;
The distance that an electron travels is called d;
When we solve for the electron's speed, we arrive at:
[tex]v=\sqrt{\frac{2 e E d}{m_e}}[/tex]
Calculate yields:
[tex]v=\sqrt{\frac{2 \cdot 1.6 \cdot 10^{-19} C \cdot 0.060 N / C \cdot 3.6 \cdot 10^{-8} \mathrm{~m}}{9.11 \cdot 10^{-31} \mathrm{~kg}}} \approx 27,5 \mathrm{~m} / \mathrm{s}[/tex]
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How did planck find the correct curve for the specturm of light emitted by a hot obkect?
Planck find the correct curve for the specturm of light emitted by a hot object by vibrational energies of the atomic resonators were quantized.
Briefing :The energy density of a black body between λ and λ + dλ is the energy E=hc/λ of a mode times the density of states for photons, times the probability that the mode is occupied. This is Planck's renowned equation for a black body's energy density. According to this, electromagnetic radiation from heated bodies emits in discrete energy units or quanta, the size of which depends on a fundamental physical constant (Planck's constant). The basis of infrared imaging is the correlation between spectral emissivity, temperature, and radiant energy, which is made possible by Planck's equation.Learn more about the Planck's constant with the help of the given link:
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He difference in speed and velocity is that a measure of velocity must include:_________
a. a destination
b. distance and time units
c. direction
d. time of departure
Considering the definition of speed and velocity, the difference is that speed has no direction while velocity does take direction into account.
Definition of speedSpeed is a quantity that expresses the relationship between the space traveled by an object.
In other words, speed can be defined as the amount of space traveled per unit of time with which a body moves .
Definition of velocityVelocity is a physical magnitude that relates the displacement of an object, the time it takes to make this change in position and direction.
Because the speed also considers the direction in which the displacement of an object occurs, it is considered a vector magnitude.
Difference between speed and velocityThe difference between speed and velocity is that speed has no direction, that is, speed is a scalar, while velocity does take direction into account, being a vectorial magnitude.
In, summary, the difference is that speed has no direction while velocity does take direction into account.
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In the system, kinetic energy from the moving water of the faucet turns the turbine. The __ energy of the spinning turbine generates __ energy which is transformed into __ energy that causes the temperature of the water to increase. The water then becomes steam and gives off more __ energy into the atmosphere.
In the system, kinetic energy from the moving water of the faucet turns the turbine. The Mechanical energy of the spinning turbine generates Electrical energy which is transformed into Thermal energy that causes the temperature of the water to increase. The water then becomes steam and gives off more Thermal energy into the atmosphere.
The mechanical energy spend on the turbine to spin would generate electrical energy. This electrical energy then converts the moving electric charge into thermal or heat energy. The heat would convert water to steam.
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What would be the average fixed cost per unit at an activity level of 5,200 units? assume that this level of activity is within the relevant range.
The average fixed cost per unit at an activity level of 5,200 units is $6.00.
What is the average fixed cost?The average fixed cost (AFC) is fixed cost that does not change with the change in the number of goods and services produced by a company. To put it in a nutshell, average fixed cost (AFC) is the fixed cost per unit and is calculated by dividing the total fixed cost by output level.
Since no cost is fixed for a long time, average fixed cost is only for a short run. When units of production increase, the average fixed cost per unit decreases. Similarly, when the business produces less units, average cost increases per unit.
However, only one unit, mostly capital, is fixed. Examples of average fixed cost are the salaries of permanent employees, mortgage payment on machinery and plant, rent, more.
Average Fixed Cost Formula:
AFC = Total fixed cost/Output (Q)
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The illustration shows a class one lever, which is being used to lift an object with a weight of 8 Newtons.
4m
What is the output force of this lever?
2 N
The output force of the given lever is determined as 32 Joules.
What is output force?
The output force of an object is the product of load or weight of an object being lifted and distance through which the load moves.
The output force of the given lever is calculated as follows;
Output force = Load x distance moved load
Output force = 8 N x 4 m
Output force = 32 Joules
Thus, the output force of the given lever is determined as 32 Joules.
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What type of boundary is depicted in the image below?
a.
transform
b.
collisional
C.
convergent
d.
divergent
Hypochlorous acid, hclo, has a pka of 7.54. at what ph are the concentration of clo- and hclo equal?
At the pH of 4.24, both the concentration of clo- and hclo are equal.
The equal concentration of clo- and hclo are 5.75 x [tex]10^{-5}[/tex] M
What is Hypochlorous acid?When chlorine dissolves in water, a weak acid called hypochlorous acid is created. This acid then partially dissociates to produce hypochlorite or ClO. Oxidizers and the main disinfectants in chlorine solutions are HClO and ClO.
It is impossible to extract HClO from these solutions because it equilibrates with chlorine very quickly.
Many commercial bleaches, deodorants, and disinfectants contain the related chemicals sodium hypochlorite (NaClO) and calcium hypochlorite (Ca(ClO)2) due to their potent antibacterial qualities.
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A train slows down as it pulls into a station. if the train is moving from your left to your right, what are the directions of the vectors for velocity and acceleration?
The directions of the vectors for velocity and acceleration are in the opposite directions.
The velocity vector is always in the direction of motion of the object. So, the direction of velocity is in the right from our point of view.When there is a positive acceleration in the object the acceleration vector is in the direction of motion of the object. When there is a negative acceleration in the object the acceleration vector is in the opposite direction of motion of the object. So, the direction of velocity is in the left from our point of view.Velocity vector is the rate of change of position of an object. Acceleration vector is the rate of change of velocity of an object.
Therefore, the directions of the vectors for velocity and acceleration are in the opposite directions.
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What tools would u bring if u were lost in forest to survive?
Explanation:
two strings fire sticks water
Answer: i would bring a flashlight, cans of food , water filter, swiss army knife, rope, and a radio
Explanation:
