Estimate the area under the curve f(x) = x^2 from x = 1 to x = 5 by using four inscribed (under the curve) rectangles. Answer to the nearest integer.
Answers
Answer:
30 square units
Step-by-step explanation:
For a rectangle to be described as "inscribed" in the context of estimating the area under a curve, the entire rectangle should be positioned underneath the curve.
Therefore, as the curve of f(x) = x² is convex (concave up) in the interval [1, 5], to estimate the area under the curve by using inscribed rectangles, we can use the Left Riemann Sum.
The Left Riemann Sum is a numerical approximation method used to estimate the definite integral of a function over a given interval by dividing the interval into subintervals. It approximates the area under the curve of a function by using rectangles, where the left side of each rectangle touches the curve at the left endpoint of each subinterval.
[tex]\boxed{\begin{minipage}{11cm}\underline{Left Riemann Sum}\\\\$\displaystyle \int^b_a f(x)\; \text{d}x \approx \Delta x \left(f(x_0)+f(x_1)+f(x_2)+...+f(x_{n-2})+f(x_{n-1})\right)$\\\\$\text{where}\; \Delta x=\dfrac{b-a}{n}$\\\end{minipage}}[/tex]
The number of subintervals, n, is the number of rectangles used, and the interval is [a, b].
As the interval is [1, 5], this means that a = 1 and b = 5.
As the number of rectangles to use is 4, then n = 4.
Calculate the value of Δx:
[tex]\Delta x=\dfrac{b-a}{n}=\dfrac{5-1}{4}=\dfrac{4}{4}=1[/tex]
The given partition divides the interval [1, 5] into 4 subintervals where the width of each subinterval is one. Therefore, the left endpoints are:
[tex]x_0=1[/tex][tex]x_1=2[/tex][tex]x_2=3[/tex][tex]x_3=4[/tex]Substitute everything into the formula and solve:
[tex]\begin{aligned}\displaystyle \int^5_1 x^2\; \text{d}x &\approx 1\cdot \left(f(1)+f(2)+f(3)+f(4))\right)\\\\&=(1)^2+(2)^2+(3)^2+(4)^2\\\\&=1+4+9+16\\\\&=30\end{aligned}[/tex]
Therefore, the estimation of the area under the curve f(x) = x² from x = 1 to x = 5 using four inscribed rectangles is 30 square units.
Related Questions
Which of the following statements is true about the triangles below?
triangleABC ≅ triangleABD by SSS
triangleABC ≅ triangleABD by ASA
triangleABC ≅ triangleABD by SAS
triangleABC ≅ triangleABD by AAA
Answers
Angle ABD = Angle ABC
Angle BAC = Angle BAD
The pair of angles that are equal lie on the side AB
Hence, triangle ABC and triangle ABD are congruent by Angle-Side-Angle (ASA)
Answer: Second option
Answer:
triangleABC ≅ triangleABD by ASA
Step-by-step explanation:
Took the test and it was correct
Fiona is serving iced tea and lemonade at a picnic. She has only 44 cups in which to serve the drinks. If x represents the number of glasses of iced tea and y represents the number of glasses of lemonade, which equation represents the number of glasses of iced tea she can serve?
Answers
Fiona can serve 22 glasses of iced tea basically.
Answer:
[tex]x=44-y[/tex]
Step-by-step explanation:
Let x represents the number of glasses of iced tea and y represents the number of glasses of lemonade.
We have been given that Fiona is serving iced tea and lemonade at a picnic. She has only 44 cups in which to serve the drinks. We can represent this information in an equation as:
[tex]x+y=44[/tex]
Since x represents the number of glasses of iced tea, so we will solve for x by subtracting y from both sides of our equation.
[tex]x+y-y=44-y[/tex]
[tex]x=44-y[/tex]
Therefore, the equation [tex]x=44-y[/tex] represents the number of glasses of iced tea Fiona can serve.
The Volume of a cube is 343ft cubed. What is the length of one side
Answers
343 = (side)^3
side = (343)^(1/3)
side = 7 ft
Rose likes to water ski on her sister's boat. the boat takes 3 gallons of gas each hour to run. gas costs $5 per gallon, how much money will she spend on gas to ski for 4 hours?
Answers
After a researcher adds 5 points to every score in a sample, the standard deviation is found to be s = 10. the original sample had a standard deviation of s = 5.
a. True
b. False
Answers
If you add 5 points to every score, that makes the difference between them 5 points higher.
A rare bacterial culture is being grown in a lab. As time passes, the cells multiply in a specific pattern.After 1 day, there is only 1 cell.After 2 days, there are 9 cells.After 3 days, there are 20 cells.After 4 days, there are 34 cells.How many cells will there be after seven days?
a.149B.120C. 98D.94
Answers
the equation is,
a_n = a_(n–1) + 3 * n + 2
Substituting the values
for day 4,
20 + 3*4 + 2 = 34
Thus the equation is correct
Therefore to calculate for a_n at n = 7 or a_7, we must calculate a_5 and a_6 first.
a_5 = 34 + 3*5 + 2 = 51
a_6 = 51 + 3*6 + 2 = 71
Hence,
a_7 = 71 + 3*7 + 2 = 94
Answer: D. 94
PLEASE HELP: Why does the AA similarity Theorem only need to identify the measurement of 2 congruent angles? Why is it not necessary to find the measurement of the third angle and show that it is also congruent to the corresponding third angle of a similar triangle?
Answers
(2h-4)11 (use the distributive property to simlify each expression)
Answers
Solve the following system of equations. Enter the y-coordinate of the solution. Round your answer to the nearest tenth. 5x+2y=7 -2x+6=9
Answers
-2x + 6y = 9
----------------
-15x - 6y = -21 (result of multiplying by -3)
-2x + 6y = 9
---------------add
-17x = - 12
x = 12/17 or 0.7
5x + 2y = 7
5(12/17) + 2y = 7
60/17 + 2y = 7
2y = 7 - 60/17
2y = 119/17 - 60/17
2y = 59/17
y = 59/17 * 1/2
y = 59/34 or 1.7 when rounded <====
Answer:
y = 1.7
Step-by-step explanation:
Given : 5x+2y=7 and - 2x+6=9.
To find : Solve the following system of equations. Enter the y-coordinate of the solution. Round your answer to the nearest tenth.
Solution : We have given
5x + 2y = 7 --------(1)
- 2x + 6y = 9 -------(2).
To make the y coefficient same multiply the equation 1 by 3.
15 x + 6y = 21 .
(+)-2x +(-) 6y = (-)9 .
______________ ( On subtracting both the equations )
17 x + 0 = 12 .
17 x = 12 .
On dividing both sides by 17
x = [tex]\frac{12}{17}[/tex].
x = 0.705
Plug x = 0.705 in equation 2 .
- 2 ( 0.705) + 6y = 9.
-1.41 + 6y = 9 .
On adding both sides by 1.41.
6y = 9 + 1.41
6 y = 10. 41
On dividing both sides by 6.
y = [tex]\frac{10.41}{6}[/tex].
y = 1.735.
Nearest tenth = 1.7
Therefore, y = 1.7.
Two coins are knocked off a table at the same time by different forces. which coin will hit the ground first?
Answers
Final answer:
Two coins knocked off a table by different forces will hit the ground simultaneously because gravity affects each coin equally, and the horizontal and vertical components of their motion are independent. This is consistent with the universality of free fall and has been empirically demonstrated by repeated experiments.
Explanation:
Do Different Forces Affect Gravity's Impact on Falling Objects?
When two coins are knocked off a table simultaneously with different horizontal forces, their subsequent motion comprises both horizontal and vertical components. However, gravity acts only on the vertical motion, and it does so equally on both coins regardless of their horizontal velocities. According to the laws of physics, specifically Galileo's theory of falling bodies, both coins will hit the floor at the same time because the horizontal and vertical motions are independent of each other. This phenomenon is due to the universality of free fall, which states that all objects accelerate at the same rate under gravity's influence, ignoring air resistance.
An important point to note is that factors such as air resistance can affect the time it takes for objects with different shapes and surface areas to hit the ground. However, in the case of the coins, since they are identical and their shapes do not change as they fall, air resistance does not play a significant role, and thus they hit the ground simultaneously. This was demonstrated by Galileo and can be observed in everyday life, such as by dropping a shoe and a coin side by side.
Marcus bought a shirt that was marked $28, but it was on sale for 15% off the marked price. what was the price of the shirt after the discount
Answers
Final answer:
To find the sale price of the shirt after a 15% discount on the marked price of $28, calculate the discount, subtract it from the original price, resulting in a sale price of $23.80.
Explanation:
Marcus bought a shirt that was marked $28, but it was on sale for 15% off the marked price. To calculate the price of the shirt after the discount, you first need to determine the amount of the discount. To do this, convert the percentage to a decimal by dividing by 100 (15% = 0.15) and then multiply this by the original price.
Discount amount = Original price × Discount rate
= $28 × 0.15
= $4.20.
Subtract the discount amount from the original price to find the sale price of the shirt:
Sale Price = Original price - Discount amount
= $28 - $4.20
= $23.80.
Therefore, the price of the shirt after the 15% discount is $23.80.
the area of a rectangular field is 6603 meter squared. if the width of the field is 71 m what is it’s length
Answers
Area of rectangular field is 6603m². Given width 71m, length is 93m (6603 ÷ 71).
To find the length of the rectangular field, you can use the formula for the area of a rectangle:
[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]
Given that the area is [tex]\( 6603 \, \text{m}^2 \)[/tex] and the width is [tex]\( 71 \, \text{m} \)[/tex], you can rearrange the formula to solve for the length:
[tex]\[ \text{Length} = \frac{\text{Area}}{\text{Width}} \][/tex]
Substitute the given values:
[tex]\[ \text{Length} = \frac{6603 \, \text{m}^2}{71 \, \text{m}} \][/tex]
[tex]\[ \text{Length} = 93 \, \text{m} \][/tex]
So, the length of the rectangular field is [tex]\( 93 \, \text{m} \).[/tex]
Three more than twice the sum of 7 and a number is 25 find the number
Answers
then you multiply 2
so now you have 3 + 2 x 7= 25
you can check it by doing 3 + 2= 5 x 7=25
Answer:
the number is 4
Step-by-step explanation:
Hello,I think I can help you with this
Step 1
convert into mathematical terms
let x the unknown number
three more: 3 +
twice the sum of 7 and a number:2(7+x)
is 25:=25
Hence
3+2(7+x)=25
Step 2
Now, solve for x
3+14+2x=25
17+2x=25
2x=25-17
2x=8
x=8/2
x=4
the number is 4
Do not hesitate to ask me if you have any concerns
Have a great day.
In the diagram below de is parallel to xy what is the value of y?
Answers
Answer:
C. 75.
Step-by-step explanation:
We have been given a diagram of two parallel lines and we are asked to find the value of y.
We can see from our given diagram that angle y and the angle that measures 105 degrees is formed inside the parallel lines and on the same side of transversal.
Since the the angles formed on the same side of transversal and inside parallel lines are called consecutive interior angles and they add up-to 180 degrees, so we can set an equation as:
[tex]y^o+105^o=180^o[/tex]
[tex]y^o+105^o-105^o=180^o-105^o[/tex]
[tex]y^o=75^o[/tex]
Therefore, the value of y is 75 degrees and option C is the correct choice.
Please help! Which direction would the graph y= Ix-4I+3 open? a) left b) up c) right d) down
Answers
because y=|x-4|+3 is y=|x-4| shifted 3 units up, entirely. So if one opens upwards, the other does so as well.
So consider the graph of y=|x-4|. This graph is the graph of y=|x| shifted 4 units right.
To show this, in the function y=|x|, consider the points (-2, 2) (because for x=-2, y=|x|=|-2|=2)
and point (2, 2).
Now consider y=|x-4|, if y = 2, then
i) x-4 = 2, which means x=6 OR
ii) x-4 =-2, which means that x=2
So we have the points (2,2) and (6, 2)
the value 2 was taken at -2 and 2 in y=|x|
the value 2 is taken at x=2 and x=6 in y=|x-4|
this proves that the graph of y=|x-4| is the graph of y=|x| shifted 4 units right.
This means that both graphs open the same way.
Thus consider the graph of y=|x|,
i) for x=0, y=0,
ii) for x>0, for larger values of x, we have larger values of y,
for example: 5>3, and |5|>3, so the graph increases
iii) for x<0, for smaller values of x, we have larger values of y
for example: -5<-3, |-5|=5>3=|-3|, thus the graph is decreasing as we move right, up to 0, then it starts increasing.
All these mean that the graph of y=|x| opens upwards, so the graph of y=|x-4|+2 also opens upwards.
Answer: b
The volume of a cube is 27n^27 cubic units. What is the length of one side of the cube?
3n^3 units
3n^9 units
27n^3g units
27n^9 units
Answers
Answer:
3n^9
this is answer peace
Suppose you are given that a=5 and 4a + b=6. What can you prove by using these statements and the substitution property?
Answers
4a + b = 6
4(5) + b = 6
20 + b = 6
b = 6 - 20
b = - 14 <=== u can prove that b = -14
Using the values given for 'a' and the equation '4a + b = 6', we applied the substitution property to prove that the value of 'b' is -14.
Given that a=5 and 4a + b=6, we can use the substitution property to find the value of b. The substitution property allows us to replace a with 5 in the second equation.
Performing the substitution, we have:
Substitute a with 5 in 4a + b = 6, giving 4(5) + b = 6.This simplifies to 20 + b = 6.To find b, subtract 20 from both sides of the equation; b = 6 - 20.Hence, b = -14.By substituting and manipulating the equation, we prove that b equals -14 when a equals 5 and 4a + b equals 6.
What is a mathematical phrase that contains numbers, variables, or operators, but no equal sign?
Answers
An expression could be,
5 + 3,
a + 4
It just cannot have an equal sign.
Describe the possible values for x and y when ∣x − y ∣ > 0. What does it mean when ∣x − y ∣ = 0? Can ∣x − y ∣ < 0? Explain your reasoning.
Answers
Hope that this helped!
The absolute value |x−y|>0 when x and y are different, |x−y|=0 when x and y are the same, and |x−y| cannot be < 0 because absolute values cannot be negative.
Explanation:The question is about absolute value, which refers to the distance of a number from zero on a number line. The notation |x−y| represents the absolute value of (x-y).
When |x−y|>0, it means that x and y are distinct from each other. For example, if x = 2 and y = 1, |x-y| equals |2-1|=1, which is indeed greater than 0. There are infinite such values for x and y as long as they are not the same.
When |x−y|=0, it implies that x and y are the same. The only way the absolute value of a number can be zero is when the number itself is zero. Therefore, this can only happen if x = y. If x and y are both 2, for example, |x-y| equals |2-2|=0.
|x−y| cannot be less than 0. The absolute value of a number can never be negative, because it represents the distance a number is from zero and distance cannot be negative.
Learn more about Absolute Values here:https://brainly.com/question/4691050
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Jamarcus's employer pays 80 percent of his medical insurance. If the insurance costs Jamarcus $20 a week, how much is his employer paying?
Answers
= 16 dollars
hope this helps . give brainliest .
thanks
Answer:
$80
Step-by-step explanation:
The total amount of his medical insurance is 100%. As such if Jamarcus's employer pays 80 percent , then the amount he pays would be equivalent to 20%.
As such, if the insurance costs Jamarcus $20 a week and the total amount is Y
then,
20% of Y = $20
0.2Y = $20
Y = $100
His employer will be paying
= 80% of $100
= $80
if f(x)=5x-2, and g(x)=2x+1, find (f+g)(x)
Answers
=5x-2+2x+1
=5x+2x-2+1
=7x-1
what is a skew line?
Answers
Skew lines are non-coplanar lines that do not intersect. For example, take a look at the image provided below. Lines m and n in the diagram do not lie on the same plane and they don't intersect. Therefore, we can say that line m is skew to line n.
When adding two numbers, such as 123 and 423, care is taken to first line them up and then add like digits. How does expanding this expression to [(1 × 10^2 ) + (2 × 10^1 ) + (3 × 10^0 )] + [(4 × 10^2 ) + (2 × 10^1 ) + (3 × 10^0 )] make the operation more like a polynomial addition problem?
Answers
Triangle FGH is an isosceles right triangle with a hypotenuse that measures 16 units. An altitude, GJ, is drawn from the right angle to the hypotenuse.
What is the length of GJ?
A. 2 units
B. 4 units
C. 6 units
D. 8 units
Answers
The value of the length GJ for the given triangle will be 8 units thus option (D) is correct.
What is a triangle?A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are three sides and three angles in every triangle, some of which may be the same.
As per the given isosceles triangle which has two equal sides so suppose the length of those sides is x unit.
By Pythagoras' theorem,
⇒ x² + x² = 16²
⇒ 2x² = 256
⇒ x² = 128
The area of a triangle is given as,
⇒ (1/2) × x × x
⇒ (1/2) x²
⇒(1/2) × 128
⇒ 64 units²
The area of the triangle will also be given as,
A = (1/2) × 16 × (GJ)
Compare both areas as
64 = (1/2) × 16 × GJ
GJ = 8 units
Hence"For the provided triangle, the length GJ value will be 8 units".
For more about triangles,
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Dan is using tiles to add –8 + 6. He begins with the tiles shown below.
mc006-1.jpg
What is the sum of –8 + 6?
–14
–2
2
14
Answers
Answer:
Step-by-step explanation:
Alright, lets get started.
The given expression is : [tex]-8+6[/tex]
-8 can be written as [tex]-6-2[/tex]
So, the expression will be = [tex]-6-2+6[/tex]
The plus six and minus six will be cancelled. so,
-2 will be remaining.
So the answer is -2 : Answer
Hope it will help :)
Answer:
-2
Step-by-step explanation:
What is the max number of slices of pizza can you get out out of 6 cuts?
Answers
So, if you make 6 cuts in a pizza, or really anything, you will get 12 pieces of it.
:)
Stephanie read 72 page on sundy and 83 pages on monday about haw many pages did stephanie read during the two days
Answers
83 pages on monday =
155 pages
There are seven vegetables. Two are squash. Two are cucumbers. The rest are carrots. You randomly choose a vegetable. Find P(cucumber). A. 1/7 B. 1/4 C. 2/7 D. 1/2
Answers
Which of the following best describes the algebraic expression 15x + 8? A. the sum of fifteen times a number and eight B. the product of fifteen plus a number and eight C. fifteen times the sum of a number and eight D. fifteen times the quantity of a number plus 8
Answers
Answer: A. [tex]\text{the sum of fifteen times a number and eight }[/tex]
Step-by-step explanation:
The given algebraic expression : [tex]15x + 8[/tex]
We can see that the constant term here is [tex]8[/tex].
The variable term is [tex]15x[/tex].
The operation used here = Addition
To convert any algebraic expression into a statement we represent the variable i.e. x by word "number".
Then the variable term is represented as : Fifteen times a number
Therefore, the best describes the given algebraic expression will be :-
"[tex]\text{the sum of fifteen times a number and eight }[/tex]"
Susie scored 85, 84,and 78 on three of the four math tests. if her average score for the four tests was 86. what was her score on the fourth test?
Answers
(247 + x) / 4 = 86
247 + x = 86 * 4
247 + x = 344
x = 344 - 247
x = 97 <=== her score was 97