Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base, estimate the area under the graph using first two and then four rectangles. f(x)equals=x squared2 between xequals=1 and xequals=2
Answers
The area under the graph by using the first two and then four rectangles is [tex]2.958[/tex] units square.
For reference use the below-given graph.
Given function is
[tex]f(x)=x^{2}[/tex] when [tex]x=1[/tex] to [tex]x=2[/tex] .
The first rectangle of the first part graph goes from [tex]1.0[/tex] to [tex]1.6[/tex], so the width will be [tex]0.6[/tex] units. And the height measured from the middle point i.e. [tex]1.3[/tex] is
[tex]f(1.3)=(1.3)^{2}[/tex]
[tex]=1.69[/tex] units.
Then the area of the first rectangle is [tex]0.6\times1.69=1.014[/tex] units square.
Similarly, the second rectangle of the first part graph goes from [tex]1.6[/tex] to [tex]2.0[/tex], so the width will be [tex]0.4[/tex] units. And the height measured from the middle point i.e. [tex]1.8[/tex] is
[tex]f(1.8)=(1.8)^{2}[/tex]
[tex]=3.24[/tex] units.
So, the area of the second rectangle is [tex]0.6\times3.24=1.944[/tex] units square.
Hence, the final area under the graph will be [tex]1.014+1.944=2.958[/tex] units square.
Further, we can do the same for another part of the graph to find the area under the graph by using four rectangles.
For example, the first rectangle of the four has a width of [tex]0.6[/tex] units and a height of [tex]f(1.1)=(1.1)^{2}[/tex]
[tex]=1.21[/tex] units.
Therefore, the area under the graph by using the first two and then four rectangles is [tex]2.958[/tex] units square.
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The estimated areas under the curve of the function f(x)=x^2 between x = 1 and x = 2 are 2.3125 using two rectangles and 2.3281 using four rectangles
Explanation:To estimate the area under the graph of the function f(x)=x^2 between x = 1 and x = 2 using rectangles, we use the method of midpoint Riemann sums. For this question, let's use 2 rectangles and then 4 rectangles.
First, for 2 rectangles, the interval from 1 to 2 is divided into 2 equal parts: [1, 1.5] and [1.5, 2]. The midpoints of these intervals are 1.25 and 1.75. The height of each rectangle is given by the function value at these midpoints: [tex]f(1.25) = (1.25)^2 =1.5625, and f(1.75) = (1.75)^2 = 3.0625.[/tex] The total area of the rectangles is thus (0.5 * 1.5625) + (0.5 * 3.0625) = 2.3125.
Next, for 4 rectangles, the interval from 1 to 2 is divided into 4 equal parts: [1, 1.25], [1.25, 1.5], [1.5, 1.75], [1.75, 2]. The midpoints of these intervals are 1.125, 1.375, 1.625, 1.875. The height of each rectangle is given by the function value at these midpoints: [tex]f(1.125) = (1.125)^2 = 1.26562, f(1.375) = (1.375)^2 = 1.8906, f(1.625) = (1.625)^2 = 2.6406[/tex], and f(1.875) = (1.875)^2 = 3.5156. The total area of the rectangles is thus [tex](0.25 * 1.26562) + (0.25 * 1.8906) + (0.25 * 2.6406) + (0.25 * 3.5156) = 2.3281.[/tex]
These are the estimated areas under the curve for 2 rectangles and 4 rectangles respectively. And as you can see, the more rectangles we use, the closer we get to the actual area under the curve.
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Related Questions
an airline ramp with a base of 50 makes an angle of 27 with the ground. find the distance a person will walk on a ramp round to the nearest foot
Answers
make sure your calculator is in Degree mode.
Which is the equation of the line with slope 5 that contains point (-2, -3)
A. y + 2 = 5(x – 3) B. y + 3 = 5(x + 2) C. y – 3 = 5(x – 2) D. y – 2 = 5(x – 3)
Answers
here's how you get it:
Start with the base y = mx + b
then imput the points and the slope:
-3 = 5 ( -2 ) + b
Then solve for b
-3 = 5 ( -2 ) + b
-3 = -10 + b
-b = -7
b = 7
so the equation is:
y = 5x + 7
The answer is y = 5x + 7
here's how you get it:
Start with the base y = mx + b
then imput the points and the slope:
-3 = 5 ( -2 ) + b
Then solve for b
-3 = 5 ( -2 ) + b
-3 = -10 + b
-b = -7
b = 7
so the equation is:
y = 5x + 7
A pea seller had some peas. He sold 40% of his stock and still had 420 peas. Originally, he had how many peas?
Answers
100-40= 60%
60% = 0.60
420/0.60 = 700
he had 700 peas
? ------------100%
100 x 420 / 60 = 700
answer
he had 700 peas originally
The table below represents the distance of a train from its destination as a function of time: Time (hours) x Distance (miles) y 0 665 1 570 2 475 3 380 4 285 Part A: What is the y-intercept of the function, and what does this tell you about the train? (4 points) Part B: Calculate the average rate of change of the function represented by the table between x = 1 to x = 4 hours, and tell what the average rate represents. (4 points) Part C: What would be the domain of the function if the train continued to travel at this rate until it reached its destination? (2 points)
Answers
B. Ave rate of change is -95
C. y=0 so x=7 Domain is [0,7]
What is the greatest common factor of 4k, 18k4, and 12?
Answers
4k=2*2*k
18k⁴=2*3*3*k*k*k*k
12=2*2*3
GCF=2
the greatest common factor is 2
You are trying to determine the half-life of a new radioactive element you have isolated. You start with 1 gram, and 5 days later you determine that it has decayed down to 0.7 grams. What is its half-life? (Round your answer to three significant digits.)
Answers
t(1/2)=[tln2]/[lnN-lnN(t)]
where;
t=time
N=initial amount
N(t)=final amount
thus the half-life of our material will be as follows:
t=5 days
N=1 g
N(t)=0.7 g
N(1/2)=[5*ln2]/[ln1-ln0.7]
=9.717
Find m2, if m6 = 57° and m7 = 23°.
Answers
Final answer:
To find m2, subtract m6 from 180 degrees.
Explanation:
To find m2 in this problem, we can use the fact that angles in a triangle sum up to 180 degrees. We know that m6 = 57° and m7 = 23°.
First, we can find m1 by subtracting m7 from 180 degrees: m1 = 180° - 23° = 157°.
Next, we can find m2 by subtracting m6 from 180 degrees: m2 = 180° - 57° = 123°. Therefore, m2 is equal to 123 degrees.
A new cruise ship line just launched 3 new ships: the Pacific Paradise, the Caribbean Paradise, and the Mediterranean Paradise.
The Caribbean Paradise has 29 more than the Pacific Paradise. The Mediterranean Paradise has 30 fewer deluxe staterooms than twice the number of deluxe staterooms on the Pacific Paradise. Find the number of deluxe staterooms for each of the ships if the total of deluxe staterooms for the total of the three ships is 719.
Answers
c = p + 29
m = 2p - 30
c + m + p = 719
(p + 29) + (2p - 30) + p = 719
4p - 1 = 719
4p = 719 + 1
4p = 720
p = 720/4
p = 180 <=== pacific paradise deluxe staterooms
c = p + 29
c = 180 + 29
c = 209 <=== caribbean paradise deluxe staterooms
m = 2p - 30
m = 2(180) - 30
m = 360 - 30
m = 330 <=== mediterranean paradise deluxe staterooms
To find the number of deluxe staterooms for each of the three ships, we can set up and solve an equation based on the given information.
Explanation:Let's represent the number of deluxe staterooms on the Pacific Paradise ship as x.
The number of deluxe staterooms on the Caribbean Paradise is 29 more than the Pacific Paradise, so it can be represented as x + 29.
The number of deluxe staterooms on the Mediterranean Paradise is 30 fewer than twice the number of deluxe staterooms on the Pacific Paradise, so it can be represented as 2x - 30.
We can form an equation with the sum of the deluxe staterooms on all three ships: x + (x + 29) + (2x - 30) = 719.
Simplifying the equation gives us 4x - 1 = 719.
By adding 1 to both sides and dividing by 4, we can find the value of x which represents the number of deluxe staterooms on the Pacific Paradise ship.
Finally, we can substitute the value of x into the expressions for the other two ships to find their number of deluxe staterooms.
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1 + 2 + 3 + 4 + 5 + . . . is an example of a finite series. True False
Answers
(Technically all series are infinite, if they are not infinite they should be referred to as sequences.)
100 - 99 + 98 - 97 + 96 ect formula
Answers
a(n)=a+d(n-1), a=initial term, d=common difference, n=term number
Since a=100 and d=-1 we have:
a(n)=100-1(n-1), which we can simplify as:
a(n)=100-n+1
a(n)=101-n
Now the only thing that we need to correct is to have the signs of the numbers alternate between positive and negative.
Since the first term is positive, we could say -1^(n+1). This expression will start out positive and will alternate back and forth from then onward. Now we simply have to multiply our sequence by this coefficient...
a(n)=-1^(n-1)*(101-n)
The above will produce the sequence you have listed....
Expand the fraction 72/125 .so that it has a denominator of 1000.
Answers
x=72000/125
x=576
So the fraction is 576/1000
Find the common ratio of the geometric sequence: −320,80,−20,5,…−320,80,−20,5,… A. −14−14 B. 1414 C. -4 D. 4
Answers
Divide each term by the previous term to get the common ratio r
r = (term 2)/(term 1) = 80/(-320) = -1/4
r = (term 3)/(term 2) = -20/80 = -1/4
r = (term 4)/(term 3) = 5/(-20) = -1/4
No matter which term you pick (as long as it's not the first one) dividing it by the previous term leads to -1/4 or -0.25 as the common ratio
Answer: -1/4
It seems like none of the answer choices have -1/4 but it could be choice A was supposed to show "-1/4" instead of "-14"? Perhaps the division sign didn't show up for some reason?
what is the remainder when 3 is synthetically divided into the polynomial -x^2+5x-9
Answers
Answer:
REmainder is -3
Step-by-step explanation:
find remainder when 3 is synthetically divided into the polynomial -x^2+5x-9
In synthetic division we write all the coefficient of the polynomial and divide by 3
3 -1 5 -9
0 -3 6
-------------------------------
-1 2 -3 ----> Remainder
So the remainder is -3
The length of a shadow of a building is
29 m
. The distance from the top of the building to the tip of the shadow is
35 m
. Find the height of the building
Answers
35^2=29^2+y^2
y^2=35^2-29^2
y^2=384
y=√384
y≈19.60 m (to the nearest hundredth of a meter, or nearest centimeter)
Which of the following is not an appropriate method of proving triangles congruent a) AAA b)SSS c) AAS D) SAS
Answers
Answer:
a) AAA.
Step-by-step explanation:
Triangles which are of the same shape and size are called congruent triangles. Triangles of same shape are similar triangles.
Two triangles can be proved congruent by SSS ,SAS AAS ,ASA ,HL postulates.
Two triangles can be proved similar by AAA method.
All congruent triangles are similar but all similar triangles are not congruent.
Hence option a) AAA is not an appropriate method of proving triangles congruent.
How many times do you need to divide by ten to get from 5947.7 to 5.9477 ?
Answers
so if u count from behind 7 to where the decimal point is it would be 4
brainliewst and thanks plz
Example
5.9477 × 10 to the third power = 5947.7
5.9477 ÷ 10 = .59477
A rectangular greenhouse has an area of 6480 square meters and a length of 120 meters. How many meters is it in width?
Answers
120 x width = 6480
So, to solve, you would divide 6480 by 120 to find the width. 6480/120 = 54.
Therefore, the width is 54.
Comment under this post or message me for further help!
W=A/L, A=6480 and L=120 so
W=6480/120 m
W=54 m
If a wheel with a radius of 80 inches spins at a rate of 50 revolutions per minute, find the approximate linear velocity in miles per hour.
Answers
EASY 5 POINTS!!! A spherical ball just fits inside a cylindrical can, 16 centimeters tall, with a diameter of 16 centimeters. Which expression gives the volume of the sphere in cubic centimeters?
Answers
formula for volume of a sphere is V=4/3 x pi x R63
if diameter = 16 than radius = 16/2 = 8
so the equation would be 4/3 x pi x 8^3
what is the measures in degrees and radians of the angle representing 7-8th of a circle
Answers
A cone has a volume of 12 cubic inches. What is the volume of a cylinder that the cone fits exactly inside of?
Answers
notice, the volume of a cylinder which is the same height(h) and radius(r) as a cone, is three times as much of that of the cone's.
Use spherical coordinates. evaluate (9 − x2 − y2) dv, where h is the solid hemisphere x2 + y2 + z2 ≤ 4, z ≥ 0.
Answers
[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}[/tex]
which gives volume element
[tex]\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta[/tex]
and so the triple integral is given by
[tex]\displaystyle\iiint_H(9-x^2-y^2)\,\mathrm dV[/tex]
[tex]=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=0}^{\varphi=\pi/2}\int_{\rho=0}^{\rho=2}(9-\rho^2\sin^2\varphi)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta[/tex]
[tex]=\displaystyle2\pi\int_{\varphi=0}^{\varphi=\pi/2}\int_{\rho=0}^{\rho=2}(9\rho^2\sin\varphi-\rho^4\sin^3\varphi)\,\mathrm d\rho\,\mathrm d\varphi[/tex]
[tex]=\dfrac{592\pi}{15}[/tex]
To solve the given task, change the Cartesian coordinates into spherical coordinates. Rewrite the integrand and the solid hemisphere limits according to these new variables. Evaluate the final triple integral using standard techniques.
Explanation:To solve the given problem, the first step is to convert the Cartesian coordinates into spherical coordinates. In spherical coordinates, a point in three dimensions is represented by three parameters: ρ (rho) for the radial distance, θ (theta) for the azimuthal angle, and φ (phi) for the polar angle.
Here, given x² + y² + z² ≤ 4, z ≥ 0, the given region corresponds to a solid hemisphere centered at the origin and radius 2 in spherical coordinates, goes as 0 ≤ ρ ≤ 2, 0 ≤ θ ≤ 2π, and 0 ≤ φ ≤ π/2. Our spherical coordinates are x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), and z = ρcos(φ).
Then, rewrite the integrand using spherical coordinates, (9 - x² - y²) becomes (9 - ρ²sin²(φ)) and dv becomes ρ²sin(φ)dρdφdθ.
The final integral that needs to be evaluated will read: ∫₀^2 ∫₀^π/2 ∫₀^2π (9 - ρ²sin²(φ))ρ²sin(φ) dθ dφ dρ. This is a triple integral in spherical coordinates that you can now solve using standard integration techniques.
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Many practical applications require calculations of area. some of these calculations are straightforward, but others are more difficult, and it turns out that the idea of the definite integral can help us with the more difficult cases. let’s start with some familiar shapes. what is the area of a rectangle of length l and width w? express your answer in terms of l and w.
Answers
Let's length be [tex]'l'[/tex] and width be [tex]'w'[/tex], then the area is
A = [tex]l[/tex] × [tex]w[/tex]
The area of a rectangle with length l and width w is given by the product l * w.
what is the area of a rectangle of length l and width w?The area of a rectangle is calculated by multiplying its length (l) by its width (w).
Therefore, the area (A) of a rectangle can be expressed as:
A = l * w
So, the area of a rectangle with length l and width w is given by the product l * w.
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The figure has ____ lines of symmetry. A. 0 B. 4 C. 6 D. 8
Answers
a 4 pointed star has 4 lines of symmetry
which algebraic expression shows the average melting points of helium, hydrogen, and neon if h represents the melting point of helium, j represents the melting point of hydrogen, and k represents the melting point of neon?
A. h + j + k
B. hjk
c. h+j+k over 3
d. hjk over 3
Answers
Answer:
c. h+j+k over 3
Step-by-step explanation:
We know that,
[tex]\text{Average}=\frac{\text{Sum of observations}}{\text{Total number of observation}}[/tex]
Here, h, j and k represents the melting point of helium, hydrogen and Neon respectively,
Since, the sum of melting points = h + j + k,
Also, the total number of gases = 3,
Thus,
[tex]\text{Average melting points}=\frac{h+j+k}{3}[/tex]
Or (h+j+k) over 3
⇒ Option c is correct.
The algebraic expression which shows the average melting
points of helium, hydrogen, and neon is h+j+k over 3.
What is Average?
This is calculated with the sum of the items divided by the
number of items.
In this scenario, we were given the items as h, j and k. The
total number of items is 3.
The average will therefore be = h+j+k/3
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Ann is grouping 38 rocks. She can put them into groups of 10 rocks or as single rocks.what are different ways Ann can group the rocks?
Answers
To answer this question, you need to make a list based on how much "10 group" of rock can be made.
The possible way to group it should be:
0 pcs of 10 group + 38 pcs of 1 group
1 pcs of 10 group + 28 pcs of 1 group
2 pcs of 10 group + 18 pcs of 1 group
3 pcs of 10 group + 8 pcs of 1 group
An indoor staircase has a 6 inch vertical rise per step to get to the second floor of the house. If the angle of elevation is 44 degrees and there are 15 feet of horizontal space total for each of the landings combined, answer the following questions showing each step of your work: a) What is the landing space per step (in inches, rounded to the nearest tenth of an inch)?
Answers
with m(A)=44°, and |AB|=15 ft.
using a scientific calculator we can find that tan 44°=0.97.
Tan(A)= (opposite side)/(adjacent side)
0.97=|CB|/15
|CB|=0.97*15=14.55 (ft) = 174.6 (in)
This means that there are 176.6/6 = 29 steps, since the height of each step is 6 in.
The total horizontal space is 15 ft = 180 in,
180/29=6.2 (in)
Answer: 6.2 in
Evaluate the limit using the appropriate limit law(s). (if an answer does not exist, enter dne.) lim x → 1 2x2 + 4 x2 + 4x − 4
Answers
[tex]\displaystyle\lim_{x\to1}\frac{2x^2+4}{x^2+4x-4}[/tex]
Since the given function is continuous at [tex]x=1[/tex], the limit of function is equal to the value of the function at [tex]x=1[/tex]:
[tex]\displaystyle\lim_{x\to1}\underbrace{\frac{2x^2+4}{x^2+4x-4}}_{f(x)}=f(1)=\frac{2+4}{1+4-4}=6[/tex]
Find the exact value of cos 105°.
Answers
find the values of x between 0 and 2pi where the tangent line to the graph of y=sinxcosx is horizontal.
Answers
[tex]\bf y=sin(x)cos(x)\impliedby \textit{now, using the product-rule} \\\\\\ \cfrac{dy}{dx}=cos(x)cos(x)-sin(x)cos(x)\implies 0=cos(x)[cos(x)-sin(x)]\\\\ -------------------------------\\\\ 0=cos(x)\implies \boxed{\measuredangle x= \begin{cases} \frac{\pi }{2}\\\\ \frac{3\pi }{2} \end{cases}}\\\\ -------------------------------\\\\[/tex]
[tex]\bf 0=cos(x)-sin(x)\implies sin(x)=cos(x)\\\\ thus\qquad \sqrt{1-cos^2(x)}=cos(x)\implies 1-cos^2(x)=cos^2(x) \\\\\\ 1=2cos^2(x)\implies \cfrac{1}{2}=cos^2(x)\implies \pm\sqrt{\cfrac{1}{2}}=cos(x) \\\\\\ \pm\cfrac{1}{\sqrt{2}}=cos(x)\implies \pm\cfrac{\sqrt{2}}{2}=cos(x)\implies \boxed{\measuredangle x= \begin{cases} \frac{\pi }{4}\\\\ \frac{3\pi }{4}\\\\ \frac{5\pi }{4}\\\\ \frac{7\pi }{4} \end{cases}}[/tex]