How do you determine the area under a curve in calculus using integrals or the limit definition of integrals?
Answers
Answer:
Please check the explanation.
Step-by-step explanation:
Let us consider
[tex]y = f(x)[/tex]
To find the area under the curve [tex]y = f(x)[/tex] between [tex]x = a[/tex] and [tex]x = b[/tex], all we need is to integrate [tex]y = f(x)[/tex] between the limits of [tex]a[/tex] and [tex]b[/tex].
For example, the area between the curve y = x² - 4 and the x-axis on an interval [2, -2] can be calculated as:
[tex]A=\int _a^b|f\left(x\right)|dx[/tex]
= [tex]\int _{-2}^2\left|x^2-4\right|dx[/tex]
[tex]\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx[/tex]
[tex]=\int _{-2}^2x^2dx-\int _{-2}^24dx[/tex]
solving
[tex]\int _{-2}^2x^2dx[/tex]
[tex]\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1[/tex]
[tex]=\left[\frac{x^{2+1}}{2+1}\right]^2_{-2}[/tex]
[tex]=\left[\frac{x^3}{3}\right]^2_{-2}[/tex]
computing the boundaries
[tex]=\frac{16}{3}[/tex]
Thus,
[tex]\int _{-2}^2x^2dx=\frac{16}{3}[/tex]
similarly solving
[tex]\int _{-2}^24dx[/tex]
[tex]\mathrm{Integral\:of\:a\:constant}:\quad \int adx=ax[/tex]
[tex]=\left[4x\right]^2_{-2}[/tex]
computing the boundaries
[tex]=16[/tex]
Thus,
[tex]\int _{-2}^24dx=16[/tex]
Therefore, the expression becomes
[tex]A=\int _a^b|f\left(x\right)|dx=\int _{-2}^2x^2dx-\int _{-2}^24dx[/tex]
[tex]=\frac{16}{3}-16[/tex]
[tex]=-\frac{32}{3}[/tex]
[tex]=-10.67[/tex] square units
Thus, the area under a curve is -10.67 square units
The area is negative because it is below the x-axis. Please check the attached figure.
Related Questions
A ladder leaning against a wall forms a triangle and exterior angles with the wall and the ground. What is the measure of exterior angle formed on the ground? 5 degrees 90 degrees 125 degrees 180 degrees
Answers
In linear equation 125 degrees is the measure of exterior angle formed on the ground.
What in mathematics is a linear equation?
There are only one or two variables in a linear equation. No variable can be multiplied by a number larger than one or used as the denominator of a fraction in a linear equation. All of the points fall on the same line when you identify the values that together make a linear equation true and plot those values on a coordinate grid.we know that the sum of all the angles of a triangle is 180
therefore
11x+7x+90=180
18x=90
x=5
exterior angle formed on the ground=180-11x
=180-55
=125
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2) Jerry and Gwen were running a relay race. The race was 12.06 kilometers total. If Jerry ran
6.76 kilometers how far did Gwen run?
3.
4.
Answers
Answer:
5.3 kilometers
Step-by-step explanation:
12.06 - 6.76 = 5.3
Solve the following quadratic equation for all values of x in simplest form.
11- x² = -3
Answers
Answer:
x = √14
Step-by-step explanation:
11- x² = -3
x² = 14
x = √14
3x+18+4x+15=180 what is x
Answers
Answer:
x = 21
Step-by-step explanation:
Adding numbers:
3x+ 18 + 4x + 15 = 180
3x + 33 + 4x = 180
Combine Like terms:
3x + 33 + 4x = 180
7x + 33 = 180
Subtract:
7x + 33 - 33 = 180-33
7x = 147
Divide:
[tex] \frac{7x}{7} = \frac{147}{7} [/tex] x = 21A manager recorded the performance review scores for each employee and placed the results in the bar chart below. All employees received a rating on each of the Evaluation Categories. If Person 6 obtained the highest score possible, what score did Person 2 receive? Use the graph and tables below.
Answers
The score of person 2 is 30.
What is a bar chart?A bar chart is a graph that shows the numerical values of variables represented by the height rectangles of equal width. In a bar graph, the rectangles must be of equal width and the space between each bars must be equal.
Total score of person 6 with the highest score possible : highest score x number of options
5 x 9 = 45
This means that on the bar chart, the value of person 6's rectangle is 45. This means that each the value of the graph increases by 5. This means that each line increases by the value of 5.
Score of person 2 = number of lines to the height of the bar x 5
6 x 5 = 30
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SOMEONE HELP WITH THIS PROBLEM
Answers
Answer:
y < 3x + 2
y> -2x + 2 The symbol should be > with a line under it to show greater than or equal to
Step-by-step explanation:
work
Question 2, 2.3.5
Part 1 of 2
Select the correct choice below and, if necessary, fill in the answer box within your choice
OA. The slope is (Simplify your answer.)
OB. The slope is undefined.
←
Find the slope of the line passing through the pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical
(7.1) and (2.1)
Answers
The slope of the line passing through the pair of points is
Given points are:
A (7,1) = (x₁ , y₁)
B (2,1) = (x₂ , y₂)
slope of a line is : y₂ ₋ y₁/x₂ ₋ x₁
Slope equals a height change /Change in width. You may also say that a slope is a climb or a run. This formula changes if the graph's horizontal axis is represented by x and the vertical axis by y.
Assume that the line's slope is'm'.
m = y₂ ₋ y₁/x₂ ₋ x₁
m= 1 ₋ 1/2 ₋ 7
m = 0/₋5
m = 0
hence, when the slope is 0 then the line is horizontal .
Therefore, the slope is 0 which indicates that the line is horizontal.
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d•r=r•d
O Commutative Property of Multiplication
O Identity Property of Multiplication.
O Zero Property of Multiplication
O Commutative Property of Addition
Answers
Answer:
Commutative property of multiplication
You need a 25% alcohol solution. On hand, you have a 325 mL of a 20% alcohol mixture.
You also have 50%
alcohol mixture. How much of the 50% mixture will you need to add to obtain the desired solution?
You will need
mL of the 50% solution
Answers
We need 65 ml of 50% mixture to obtain the desired solution.
Here, we need to make a 25% alcohol solution by mixing some amount of 50% alcohol mixture with 325 mL of a 20% alcohol mixture.
Let the amount of 50% alcohol solution be x
The the quantity of alcohol in 50% alcohol solution = 50% of x
= 0.5x
Similarly the the quantity of alcohol in 20% alcohol solution = 20% of 325
= 65ml
The total quantity of 25% alcohol solution = (325 + x) ml
Thus, the the quantity of alcohol in 25% alcohol solution = 25% of (325 + x)
= 0.25 (325 + x)
= 81.25 + 0.25x
Thus, we can form the following equation-
0.5x + 65 = 81.25 + 0.25x
0.5x - 0.25x = 81.25 - 65
0.25x = 16.25
x = 16.25/ 0.25
x = 65
Thus, we need 65 ml of 50% mixture to obtain the desired solution.
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A bag contains 10 marbles: 5 are green, 3 are red, and 2 are blue. Deandre chooses a marble at random, and without putting it back, chooses another one at random. What is the probability that the first marble is red and the second is blue? Write your answer as a fraction in simplest form.
Answers
Answer:
1/15
Step-by-step explanation:
3/10 x 2/9 = /90 Dived the top and bottom by 6 and you will get 1/15.
The first time there were 10 marbles and she wanted red. There were only 3 red so the chances of getting a red is 3/10. Now she only has 9 marbles left. She was a blue so her chances are 2/9.
Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.
j2 + 20j +
Please explain in detail.
Answers
Answer:
the quadratic equations is 22+×3÷78=36-54Stanley can purchas a 15.4-ounce bottle of
olive oil for $5.39 or a 23.6-ounce bottle of olive oil for $7.08.
Which bottle is the better buy? Explain.
Answers
Answer: They are the same deal
Step-by-step explanation:
15.4/5.39=3.33333...
23.6/7.08=3.333333...
Can someone do 4,6,8 for me pls show work I give l Brainly and extra point
Answers
Answer:
4.) T = M - N
T(-M) = M - N (-M)
(/-1) T - M = -N (/-1)
-T + M = N
6.) Ax + By = C
(-Ax) Ax + By = C (-Ax)
By = C - Ax
By (/B) = (C - Ax)/(B)
y= (C - Ax)/B
8.) C = pi(d)
(/pi) C = d
C/pi = d
A transportation company charges a fee of $125 plus $50 per hour to rent a limousine. If John paid $425 for a limousine, for how many hours did he rent the limousine?
A. 7
B. 6
C. 5
D. 4
Answers
John rented the limousine for 6 hours.
The minimum fee charged by the transportation company to rent a limousine is $125.The hourly fee charged by the transportation company to rent a limousine is $50.The amount of money paid by John to rent a limousine is $425.Let the number of hours for which John paid for the rent of the limousine be "x".A formula that represents the equality of two expressions by linking them with the equals sign is known as an equation.The above-given data helps us to form the linear equation given below :425 = 125 + 50x300 = 50xx = 6Thus, John rented the limousine for 6 hours.To learn more about equations, visit :
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At a price of $180 there is demand for 2520 items and a supply of 1440 items. At a price of $330 there is demand for 1470 items and a supply of 2640 items. Assuming supply and demand are linear, find the equilibrium price and quantity.
Answers
Assuming supply and demand are linear, the equilibrium price is equal to $252 and the equilibrium quantity is equal to 2,016 items.
What is a linear function?A linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.
First of all, we would determine a linear function for both supply and demand by finding the slope using this formula as follows:
[tex]Slope,\;m = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope,\;m = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]
Note: price (p) in dollars would be the output and quantity (q) would be the input.
For supply, the points include the following:
Points (p, q) = (1,440, 180).Points (p, q) = (2640, 330).Slope, m = (330 - 180)/(2640 - 1440)
Slope, m = 150/1200
Slope, m = 1/8.
Next, we would find the y-intercept at (1,440, 180):
p = qm + c
180 = 1/8(1440) + c
180 = 180 + c
c = 0.
Therefore, the supply equation is p = q/8.
For demand, the points include the following:
Points (p, q) = (2,520, 180).Points (p, q) = (1,470, 330).Slope, m = (330 - 180)/(1,470 - 2,520)
Slope, m = 150/-1,050
Slope, m = -1/7.
Next, we would find the y-intercept at (2,520, 180):
p = qm + c
180 = -1/7(2,520) + c
180 = -360 + c
c = 180 + 360.
y-intercept, c = 540.
Therefore, the demand equation is p = -q/7 + 540.
In order to find the equilibrium price and quantity, we would set the supply equal to the demand:
q/8 = -q/7 + 540
Multiplying all through by 56, we have:
7q = -8q + 30,240
15q = 30,240
q = 30,240/15
Equilibrium quantity, q = 2,016 items.
For the equilibrium price, we have:
p = q/8
p = 2,016/8
Equilibrium price, p = $252.
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What is the MEDIAN of this set of data?
10, 90, 80, 60, 50, 40, 70, 30, 20, 100
Answers
Answer:
your answer is 55 as shown by above photo ^^
Answer:
55
Step-by-step explanation:
The set of real numbers is composed of rational and irrational numbers.
A. true
B.false
Answers
Answer:
true..... ............
Simplify the expression (X1/6)3
X2
X3
X1/3
X1/2
Answers
What is the domain of the function y = 2√x-5?
Answers
THE ANSWER IS IN THE PHOTO
Read this excerpt from “The All American Slurp”
In china we never ate celery raw or any other kind of vegetable raw. We always had to disinfect the vegetables in boiling water first. When we were presented with our first relish tray, the raw celery caught us unprepared
What story element does the author develop in this excerpt?
Conflict between family members.
Character traits of the Lin family.
Plot details about the Gleason party.
Setting details about the Lins hometown.
Answers
From the excerpt that we have here the story element that the Character traits of the Lin family.
What does the character trait mean?This is the term that is used to refer to the words that are used in order to offer the characterization of a person. These are the terms that would help to show the personalities and the qualities that a person has. It tells us who these persons are and the way they would be described if we are to talk about them to other people.
Based on the description that I have given here, the way that we would be able to talk about this family would be that they are known to prepare their food in a different way that the Americans made theirs.
From what they said, they would always have to put their vegetables first in boiling water. This would make it less raw. But from what they were offered in America, they had to eat the vegetables that had been left raw.
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Just 17 and 18 please!
Answers
[16] f(-3) = -5
#17 is asking us to find what the function equals when x is equal to -3. We will find x = -3 on the graph, then find what y equals.
[17] f(4) = 0
#17 is quite similar, it is asking us to find what the function equals when x is equal to 4. We will find x = 4 on the graph, then find what y equals.
You asked in the comments for #16 and #17, I hope this helps.
the difference of twice
a number and seven is nine
Answers
Answer:
16!
Step-by-step explanation:
That the number squaring is 8
A chemical company makes two brands of antifreeze. The first brand is 20% pure antifreeze, and the second brand is 95% pure antifreeze. In order to obtain 140 gallons of a mixture that contains 65% pure antifreeze. How many gallons of each brand of antifreeze must be used?
Answers
If a chemical company makes two brands of antifreeze. The first brand is 20% pure antifreeze, and the second brand is 95% pure antifreeze. The number of gallons each brand of antifreeze must be used is: 56 gallons of 20% antifreeze and 84 gallons of 95% antifreeze.
Number of gallons for each brand of antifreezeLet call x = The amount of 20% pure antifreeze needed
Let 140 - x = amount of 95% antifreeze needed
Hence,
x gallons of 20% antifreeze + (140-x) gallons of 95% = 140 gallons of 65%
Thus,
.20x + .95(140-x) = .65(140)
.20x + 133 - .95x = 91
Collect like terms
42= .75x
Divide both side by .75x
x=42/.75
x=56
Hence,
56 gallons of 20% antifreeze
84 gallons of 95% antifreeze
Therefore the number of gallons each brand of antifreeze must be used is: 56 gallons of 20% antifreeze and 84 gallons of 95% antifreeze.
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f(x) = cube root 2x
g(x) = 2x + 1
Find (f/g) (X) Include any restrictions on the domain
Answers
This might be help
Recall the equation for a circle with center ( h , k ) and radius r . At what point in the first quadrant does the line with equation y = 0.5 x + 2 intersect the circle with radius 5 and center (0, 2)?
Answers
At (4.472 , 4.236) in the first quadrant , the line with equation y = 0.5 x + 2 intersect the circle with radius 5 and center (0, 2) .
The equation of circle with center (h,k) and radius r can be written as
(x-h)²+(y-k)²=r²
In the given question
the center of the circle is (0,2) and the radius of the circle is 5 .
So the equation of circle become
(x-0)²+(y-2)²=5²
x²+(y-2)²=25 ...(i)
It is given that y=0.5x+2 intersects the circle
y = 0.5x+2
Rewriting the equation we get
y-2=0.5x
(y-2)=0.5x
x=(y-2)/0.5
x = 2(y-2) ...(as 0.5=1/2) ....(ii)
Squaring both sides
x² = (2(y-2))²
x² = 2²(y-2)²
x² = 4(y-2)² ...(iii)
Substituting the value of equation (iii) in equation (i) we get ,
4(y-2)²+(y-2)²=25
5(y-2)²=25
(y-2)²=25/5
(y-2)²=5
Taking Square root both the sides , we get
√(y-2)²=√5
y-2 = √5
y=(√5)+2
y=2.236+2
y=4.236
Substituting the value of y in (ii) we get
x=2(4.236-2)
x=2*2.236
x=4.472
Therefore , the circle with radius 5 and center (0, 2) will intersect the line with equation y = 0.5 x + 2 in the first quadrant at (4.472 , 4.236).
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A side of the triangle below has been extended to form an exterior angle of 125. Find the value of x
Answers
Answer:
55°
Step-by-step explanation:
The angle that is represented with x and the exterior angle with a measure 125° are supplementary angles.
As known the sum of supplementary angles is 180 so the value of x:
180 - 125 = 55 and
x = 55 is the answer to the question.
A bathtub is partially filled with g gallons of water.
Alice adds 10 more gallons of water to the tub,
making it 50% full. If Marco then adds 40 more
gallons of water, the tub will be 100% full. What is
the value of g?
A. 20
B. 24
C. 30
D. 40
Answers
Answer: C.) 30
Step-by-step explanation: 40 gallons = 50% full Alice added 10 more gallons to make it 50% full there was already 30 gallons in.
Divide 6/13 by 6/12 .
A. 12/13
B. 9/16
C. 1/12
D. 13/12
Answers
Which of the following sets is NOT closed under addition.
A) Natural Numbers
B) Whole numbers
C) Integers
D) Odd numbers
Answers
The odd numbers do not follow the closure property of addition.
What is a closure property of addition?
The closure property of addition states that when any two numbers from the set of reals are added, the result will always be a real number. For example 2 + 3 = 5
Explanation
When we add any two natural numbers, the result is always a natural number, i.e. 2 + 1 = 3. Thus, it follows the closure property of addition.
When we add any two whole numbers, the result is always a whole number, i.e. 3 + 0 = 3. Thus, it follows the closure property of addition.
When we add any two integers, the result is always an integer, i.e. 2 + (-5) = -3. Thus, it follows the closure property of addition.
When we add any two odd numbers, the result is not an odd number, i.e. 3 + 1 = 4. Thus, it does not follow the closure property of addition.
Hence, the odd numbers do not follow the closure property of addition.
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Which of the following properties is demonstrated by the equation below?
9(x+3)=9x+27
distributive property
inverse property of addition
associative property of addition
communitive property of addition
Answers
Answer: Distributive Property
Step-by-step explanation:
This is an example of the distributive property because the 9 is being distributed into each term, yielding the 9x and the 27.