Find the area of the region enclosed by the graphs of the functions
f(x)=x, g(x)=x^3
by partitioning the x- axis.
(Use symbolic notation and fractions where needed.)
Answers
Answer:
[tex]\displaystyle A = \frac{1}{2}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightEquality PropertiesAlgebra I
Terms/CoefficientsFunctionsFunction NotationGraphingCalculus
Area - Integrals
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Area of a Region Formula: [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
Step-by-step explanation:
*Note:
Remember that for the Area of a Region, it is top function minus bottom function.
Also remember that finding area and evaluating are two different things.
Step 1: Define
f(x) = x
g(x) = x³
Bounded (Partitioned) by x-axis
Step 2: Identify Bounds of Integration
Find where the functions intersect (x-values) to determine the bounds of integration.
Simply graph the functions to see where the functions intersect (See Graph Attachment).
Interval: [-1, 1]
1st Integral: [-1, 0]
2nd Integral: [0, 1]
Step 3: Find Area of Region
Integration.
Substitute in variables [Area of a Region Formula]: [tex]\displaystyle A = \int\limits^0_{-1} {[x^3 - x]} \, dx + \int\limits^1_0 {[x - x^3]} \, dx[/tex][Area] Rewrite Integrals [Integration Property - Subtraction]: [tex]\displaystyle A = (\int\limits^0_{-1} {x^3} \, dx - \int\limits^0_{-1} {x} \, dx) + (\int\limits^1_0 {x} \, dx - \int\limits^1_0 {x^3} \, dx)[/tex][Area] [Integrals] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle A = [\frac{x^4}{4} \bigg|\limits^0_{-1} - (\frac{x^2}{2}) \bigg|\limits^0_{-1}]+ [\frac{x^2}{2} \bigg|\limits^1_0 - (\frac{x^4}{4}) \bigg|\limits^1_0][/tex][Area] Evaluate [Integration Rule - FTC 1]: [tex]\displaystyle A = [\frac{-1}{4} - (\frac{-1}{2})] + [\frac{1}{2} - \frac{1}{4}][/tex][Area] [Brackets] Add/Subtract: [tex]\displaystyle A = \frac{1}{4} + \frac{1}{4}[/tex][Area] Add: [tex]\displaystyle A = \frac{1}{2}[/tex]Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Area Under the Curve - Area of a Region (Integration)
Book: College Calculus 10e
Related Questions
h (5x)=2x^2 - 5
find h(5x)
Answers
Answer:
5x
=5*x
=x+x+x+x+x
=5x
Step-by-step explanation:
=5*x
=x+x+x+x+x
=5x
There are 37 students on a bus the bus stops 5 times. 7 students get off at each stop how many students are left on the bus
Answers
37-7-7-7-7-7=2
or 7x5=35 so 37-35=2
An investment of $500 increases at a rate of 12.5% per year. What is the value of the investment after 30 years? Round to the nearest dollar.
Answers
Answer:
Here's the equation:
0.125(500) + 500
But because it is 30 years:
30(0.125(500)) + 500
3.75(500) + 500
We can make it simpler:
4.75(500)
2375
You will have $2375
Step-by-step explanation:
Which expression is equivalent to (73)-2
1/7.7.7.7.7.7
7
1/7
-1/7.7.7.7.7.7
Answers
Answer:
73*-2 =-146 you have to multiply the 73 to the -2
Step-by-step explanation:
Ben loves Thanksgiving and he cooks the family turkey every year. He has noticed that over the years the amount of people who show up for his Thanksgiving dinner tends to vary, and he was wondering if this was correlated with the size of the turkeys he cooks. He recorded the weight of the turkeys he cooked (in lbs.) and the amount of guests who showed up for diner for 12 randomly chosen Thanksgivings. When he was analyzing the data he decided to perform a Box Cox transformation and he derived a lambda equal to 0.891.
What transformation does this value of lambda suggest?
a. With a lambda close to 1. it looks like we should take the square root of y
b. With a lambda close to 1. it does not look like we need to re-express the data.
c. With a lambda close to 1, it looks like we should take the log of y
d. With a lambda close to 1. it looks like we should take the log of x.
Answers
Answer:
The answer is "Option b".
Step-by-step explanation:
Calculating the common Box-Cox Transformation:
[tex]\lambda \ value \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Transformed \ data \ (Y')\\\\[/tex]
[tex]-3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{-3} =\frac{1}{Y^3} \\\\ -2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{-2} =\frac{1}{Y^2} \\\\ -1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{-1} =\frac{1}{Y^1} \\\\ -0.5 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{-0.5} =\frac{1}{\sqrt{Y}} \\\\ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \log(Y)^{**} \\\\[/tex]
[tex]0.5 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{-0.5} =\sqrt{(Y)} \\\\ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{1} =Y \\\\ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{2} \\\\ 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{3} \\\\[/tex]
13x = 17 divided by 13
Answers
Answer:
x=17/13
Step-by-step explanation:
pretty sure you just shift x alone by itself
Answer: [tex]17/13=1.30769231[/tex]
Step-by-step explanation:
x² + 3x -4 = 0
Factor the quadratic equation
Answers
Answer:
(x-1)(x+4)
Step-by-step explanation:
[tex]x^2+3x-4\\x^2-x+4x-4\\x(x-1) 4(x-1)\\(x-1)(x+4)[/tex]
First, find the factors of the constant (number without an x)
The factors of -4 are:
-1, 4
1, -4
2, -2
Looking at these factors find which pair of factors adds up to the middle number (3x)
4 + -1 = 3
Now you can factor this equation. Put x + 4 into one pair of parenthesis and x - 1 into another pair of parenthesis. You do this because you are separating the two factors into two expressions, that when multiplied together, will equal x^2 + 3x - 4
(x+4)(x-1) is the answer
To check this, multiply the two expressions together. You should get x^2+3x-4=0
A square picture frame with each side length represented by 3x -
5 has a perimeter of 52 inches. Which equation models this situation?
A. 3x - 5 + 4 = 52
B. 3x - 5 = 52
C. 4(3x + 5) = 52
D. 4(3x - 5) = 52
Answers
Answer:
D. 4(3x-5) = 52
Step-by-step explanation:
52 is equal the sum of one side multiplied by 4
So
4(3x-5) = 52
Is the right answer
decoration of line passing through PT (3 ,2) close and perpendicular to line y is equals to X is
Answers
Answer:
y = -x+5
Step-by-step explanation:
The equation of the line in point slope form is expressed as;
y - y0 = m(x-x0)
Get the slope
Given the equation of the line as y = x
The slope of the line is m = 1
The slope of the line perpendicular =M =-1/1
M = -1
Using the slope M = -1 and the point (3, 2) to get the required equation;
y - y0 = m(x-x0)
y - 2 = -1(x-3)
y - 2 = -x+3
y+x = 3 + 2
y = -x + 5
Hence the required equation is y = -x+5
Find the volume 11yd 6yd 5yd
Answers
Answer:
(Base)Height
(L*w) H
(5*6)11
(30)11
330u^3
Step-by-step explanation:
Formula, substitution, and answer as shown above
A seventh-grade student forgets to bring his lunch money on several occasions. Each time he forgets his lunch money, he has to charge his cafeteria account by .15.
Solve the following equation.
4(−3.15)=
Explain what this product means in terms of the cafeteria account.
Answers
Answer:
-12.6
Step-by-step explanation:
Which of the following is an equation of the line with slope 5 that passes through the
point (1, - 4)
Answers
Answer:
y=5x-9
Step-by-step explanation:
You want to find the equation for a line that passes through the point (1,-4) and has a slope of 5.
First of all, remember what the equation of a line is:
y = mx+b
Where:
m is the slope, and
b is the y-intercept
To start, you know what m is; it's just the slope, which you said was 5. So you can right away fill in the equation for a line somewhat to read:
y=5x+b.
Now, what about b, the y-intercept?
To find b, think about what your (x,y) point means:
(1,-4). When x of the line is 1, y of the line must be -4.
Because you said the line passes through this point, right?
Now, look at our line's equation so far: . b is what we want, the 5 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the the point (1,-4).
So, why not plug in for x the number 1 and for y the number -4? This will allow us to solve for b for the particular line that passes through the point you gave!.
(1,-4). y=mx+b or -4=5 × 1+b, or solving for b: b=-4-(5)(1). b=-9.
The equation of the line that passes through the point (1,-4) with a slope of 5
is
y=5x-9
In a few sentences, describe what this weather map tells you about the weather.
a weather map of Florida with an L in northern Florida and an H in southern Florida
Answers
The weather map of Florida with an L in northern Florida and an H in southern Florida indicates that there is a low-pressure system in the north and a high-pressure system in the south.
What is an expression?One mathematical expression makes up a term. It might be a single variable (a letter), a single number (positive or negative), or a number of variables multiplied but never added or subtracted. Variables in certain words have a number in front of them. A coefficient is a number used before a phrase.
Given:
In a few sentences, describe what this weather map tells you about the weather.
The map is given in the attached image.
The weather map of Florida with an L in northern Florida and an H in southern Florida indicates that there is a low-pressure system in the north and a high-pressure system in the south.
This pattern could result in potentially cooler temperatures, increased chances of precipitation, and cloudy skies in the north, while the south may experience warmer temperatures, clear skies, and drier weather conditions.
Therefore, the required meaning of L and H is given above.
To learn more about the expression;
brainly.com/question/24242989
#SPJ1
You randomly select ONE card from a 52-card deck. Find the probability of selecting a card greater than 2 and less than 7, or a diamond.
Answers
There is 13 diamonds and 4 numbers greater than 2 but less than 7. This means your probability of selecting one of these specific cards is a 17/52 chance
A cookie recipe that calls for 2 2/3 cups of sugar will make 32 cookies however you only have 1 2/3 cup of sugar and you have to adjust your recipe what fraction of the recipe
Answers
Explanation is in the file
tinyurl.com/wpazsebu
Stable cost reporting in a manufacturing setting is typically a sign that operations are running smoothly. The accounting department at Rockwell Collins, an avionics manufacturer, analyzes the variance of the weekly costs reported by two of its production departments. A sample of 16 cost reports for each of the two departments shows cost variances of 2.3 and 5.4, respectively. Is this sample sufficient to conclude that the two production departments differ in terms of unit cost variance? Use α = 0.10 (to 2 decimals).
Answers
Answer:
Following are the responces to the given question:
Step-by-step explanation:
For this issue the statistical programme performance is:
Two overview hypothesis test sample variance:
[tex]\to \sigma_1^2[/tex] =Population variance 1
[tex]\to \sigma_2^2[/tex] = Population variance 2
[tex]\to \frac{\sigma_1^2}{\sigma_2^2}=[/tex] Two variance ratios
[tex]\to H_0: \frac{\sigma_1^2}{\sigma_1^2}=1\\\\\to H_A: \frac{\sigma_1^2}{\sigma_1^2}\neq 1[/tex]
Following are the result of the Hypothesis test: [tex]Ratio \ \ \ \ \ \ \ \ \ \ Num.DF \ \ \ \ \ \ \ \ \ \ Den.DF \ \ \ \ \ \ \ \ \ Sample\ \ Ratio\ \ \ \ \ \ F-Stat \ \ \ \ \ \ \ P-value\\\\[/tex]
[tex]\frac{\sigma_1^2}{\sigma_1^2} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.3478261 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.3478261 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1091[/tex]
Test statistic [tex]= 2.3478 \approx 2.35[/tex]
P-value > 0.10
The null hypothesis should not be rejected.
Solve: 9x^2-49=0
A. -9/49, 9/49
B. -49/9, 49/9
C. -3/7, 3/7
D. -7/3, 7/3
Answers
Answer:
D. -7/3, 7/3
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityAlgebra I
Multiple RootsStep-by-step explanation:
Step 1: Define
9x² - 49 = 0
Step 2: Solve for x
[Addition Property of Equality] Isolate x term: 9x² = 49[Division Property of Equality] Isolate x term: x² = 49/9[Equality Property] Isolate x: x = ±7/3Help meeee.. I’ll mark you brainlist and Uc you scam then i report lol but helpppp me
Answers
Answer:
70 units²
Step-by-step explanation:
Area of the figure = area of rectangle + area of triangle
= L*W + ½*bh
L = 10
W = 5
b = 5
h = 18 - 10 = 8
Substitute:
Area of the figure = 10*5 + ½*5*8
= 50 + 20
= 70 units²
The bases of a trapezoid are 8 yards and 16 feet. The height of the trapezoid is 12 feet. What is area of the trapezoid in square feet?
Answers
Answer:
240 sq ft
Step-by-step explanation:
First turn the 8 yards to feet my multiplying by 3. You get 24.
Then
Add the bases together to get 40 Divide them by the number of bases (2) to get 20
Multiply the 20 by the height, (12) to get 240
240 sq feet
Mark me as brainliest if this helps!
Choose the equation for the line in slope-intercept form.
Answers
Leslie placed this ad in Collector Car Monthly. 1957 Chevrolet Nomad station wagon. Tropical Turquoise, 6 cyl. auto, PS, PW, AM/FM, repainted, rebuilt transmission, restored two-tone interior. Mint! Moving, sacrifice, $52,900. 555-4231
Answers
Complete Question:
Leslie placed this ad in Collector Car Monthly.
1957 Chevrolet Nomad station wagon. Tropical
Turquoise, 6 cyl. auto, PS, PW, AM/FM, repainted,
rebuilt transmission, restored two-tone interior.
Mint! Moving, sacrifice, $52,900. 555-4231
(a) If the publication charges $48 for the first three lines and $5 for each extra line, how much will this ad cost Leslie?
(b) b. Ruth buys the car for 8% less than the advertised price. How much does she pay?
(c) Ruth must pay her state 6% sales tax on the sale. How much must she pay in sales tax?
Answer:
[tex]Total = \$53[/tex]
[tex]Ruth = \$48668[/tex]
[tex]Sales\ Tax = \$2920.08[/tex]
Step-by-step explanation:
Given
[tex]Ad = 4\ lines[/tex]
[tex]Charges: \$48[/tex] --- First 3 lines
[tex]Additional: \$5[/tex] -- Extra lines
Solving (a) : Determine the total charge
The ad has 4 lines. This means that 3 of the 4 lines will be charged at $48 (in total) while the extra line will be charged at $5 per line
So, the total charge is calculated as:
[tex]Total = 3\ lines + 1\ line[/tex]
[tex]Total = \$48 + 1 * \$5[/tex]
[tex]Total = \$48 + \$5[/tex]
[tex]Total = \$53[/tex]
Solving (b): How much does Ruth pay?
From the ads, the price of the car is:
[tex]Price = \$52,900[/tex]
And Ruth pays 8% less.
This means that she pays 92% (i.e. 100% - 8% = 92%)
So, the calculation is:
[tex]Ruth = 92\% * Price[/tex]
[tex]Ruth = 92\% * \$52900[/tex]
[tex]Ruth = \$48668[/tex]
Solving (c): The sales tax
[tex]Sales\ Tax = Sales\ Tax\% * Amount\ Paid[/tex]
[tex]Sales\ Tax = 6\% * \$48668[/tex]
[tex]Sales\ Tax = \$2920.08[/tex]
A zoo requires that 1 adult accompany every 7 students that visit the zoo. How many adults must accompany 28 students?
Answers
Take 28/7 and you get 4
Answer:
4 adaults
Step-by-step explanation:
Maria bought a bag of mixed fruit that is 40% oranges, 33% green apples, and 27% red apples, If she
randomly selects 5 pieces of fruit from the
bag, how many would you expect to be oranges?
A 1
B 2
C 3
D 4
Answers
Explanation
Answer:
b
Step-by-step explanation:
please help me.. i would really appreciate it, thank you!
Answers
4x + 3 − 23x
This expression has 3 terms.
true or false
Answers
Answer:
True
Terms are separated by the signs
i need help with this problem
Answers
9514 1404 393
Answer:
(a) 4 in²
Step-by-step explanation:
The volume of a rectangular prism is given by the formula ...
V = Bh
where B is the area of the base, and h is the height of the prism.
We can find the area of the base by solving for B.
B = V/h
B = (12 in³)/(3 in)
B = 4 in² . . . . . . . . . . matches the first choice
Which of the following uses scientific notation
incorrectly?
(View picture)
Answers
G: two places before the decimal when there should only be one.
H: there is no decimal.
A box contains 4 red marbles, 9 white marbles, and 8 blue marbles. If a marble is randomly selected from the box, what is the probability that it is blue? Express your answer as a fraction or a decimal number rounded to three decimal places, if necessary.
Answers
Answer:
8/21
Step-by-step explanation:
Add the number of marbles : 4+9+8= 21
Then how many are blue: 8
Answer 8/21. Can't be simplified
0.38
Kailee and Leanne are working on their math homework. There is a question that says –2 < –4. Kailee says that that is a true statement, and Leanne said it is not true. Which girl is correct? Use the number line below to prove your answer.
Answers
What is the median of this data set 2 2 5 6 6 8 9 11?
A) 6
B)5
C)7
D) 8
Answers
Answer:
6 is your answer since its in the middle.
Step-by-step explanation: