Consider the following functions. (See attachment)
Find the area of the region.
Answers
Answer:
Area: [tex]\displaystyle \frac{1}{2}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
FunctionsFunction NotationGraphing FunctionsExponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Calculus
Derivatives
Derivative Notation
Derivative of a constant is 0
Area - Integrals
Area under a curveArea between 2 curvesIntegration 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]
U-Substitution
Area of a Region Formula: [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
Identify functions
[tex]\displaystyle f(x) = \sqrt[3]{x - 9}[/tex]
[tex]\displaystyle g(x) = x - 9[/tex]
Step 2: Identify Info
Graph the functions - See Attachment
[1st Integral] Bounds: [8, 9], g(x) top function/f(x) bottom function
[2nd Integral] Bounds: [9, 10], f(x) top function/g(x) bottom function
Step 3: Find Area Pt. 1
Set up integrals [Area of a Region Formula]: [tex]\displaystyle A = \int\limits^9_8 {[(x - 9) - \sqrt[3]{x - 9}]} \, dx + \int\limits^{10}_9 {[\sqrt[3]{x - 9} - (x - 9)]} \, dx[/tex]Rewrite integrals [Integration Property - Addition/Subtraction]: [tex]\displaystyle A = \int\limits^9_8 {(x - 9)} \, dx - \int\limits^9_8 {\sqrt[3]{x - 9}} \, dx + \int\limits^{10}_9 {\sqrt[3]{x - 9}} \, dx - \int\limits^{10}_9 {(x - 9)} \, dx[/tex]Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle A = (x^2 - 9x) \bigg| \limits^9_8 - \int\limits^9_8 {\sqrt[3]{x - 9}} \, dx + \int\limits^{10}_9 {\sqrt[3]{x - 9}} \, dx - (x^2 - 9x) \bigg| \limits^9_8[/tex]Evaluate [Integration Rule - FTC 1]: [tex]\displaystyle A = \frac{-1}{2} - \int\limits^9_8 {\sqrt[3]{x - 9}} \, dx + \int\limits^{10}_9 {\sqrt[3]{x - 9}} \, dx - \frac{1}{2}[/tex]Subtract: [tex]\displaystyle A = -\int\limits^9_8 {\sqrt[3]{x - 9}} \, dx + \int\limits^{10}_9 {\sqrt[3]{x - 9}} \, dx - 1[/tex]Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle A = -\int\limits^9_8 {(x - 9)^{\frac{1}{3}}} \, dx + \int\limits^{10}_9 {(x - 9)^{\frac{1}{3}}} \, dx - 1[/tex]Step 4: Identify Variables
Identify variables for u-substitution.
u = x - 9
du = dx
Step 5: Find Area Pt. 2
[Integrals] U-Substitution: [tex]\displaystyle A = -\int\limits^0_{-1} {u^{\frac{1}{3}}} \, du + \int\limits^1_0 {u^{\frac{1}{3}}} \, du - 1[/tex][Integrals] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle A = -(\frac{3}{4}u^{\frac{4}{3}}) \bigg| \limits^0_{-1} + (\frac{3}{4}u^{\frac{4}{3}}) \bigg| \limits^1_0 - 1[/tex]Evaluate [Integration Rule - FTC 1]: [tex]\displaystyle A = -(\frac{-3}{4}) + \frac{3}{4} - 1[/tex]Simplify: [tex]\displaystyle A = \frac{1}{2}[/tex]Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Integrals - Area between 2 curves
Book: College Calculus 10e
Related Questions
Find the missing length
Answers
Answer:
Last one
Step-by-step explanation:
solve the following :
7+2x/3= 5
and
3x+4/2= 9.5
Answers
Answer:
x= 4
x= 5
Step-by-step explanation:
7 + 2(4)/ 3 = 5
7 + 2(4)/3 = 5
7+8/3 = 5
15/3 = 5
3(5) +4/ 2 = 9.5
15+4/2 = 9.5
19/2 = 9.5
1. a box contains 11 pink 8 blues and 9 orange marbles. If you pick two marbles without looking , what is the probability that the first will be orange and the second will be pink?
2. You accidentally dropped a coin from the top of 10 stairs. What is the probability that it will land heads up on the seventh step?
Answers
2. the probability is about 47%
What is the surface area of a cereal box with a height of 10 inches, a width of 4 ½ inches, and a length of 7 inches?
Answers
Answer:
Surface area of a box = 293 inches ²
Step-by-step explanation:
Surface area of a box = 2(lw + lh + hw)
Where,
l = length = 7 inches
w =width = 4 1/2 inches
h = height = 10 inches
Surface area of a box = 2(lw + lh + hw)
= 2(7*4.5 + 7*10 + 10*4.5)
= 2(31.5 + 70 + 45)
= 2(146.5)
= 293 inches ²
Surface area of a box = 293 inches ²
please help!!
The Pythagorean Identity states that:
(sin x)2 + (cos x)2 = 1
Given cos 0 = 9/10, find sin 0.
sin 0 = [ ? ] / [ ? ]
Simplify the fraction
Answers
Answer:
[tex] sin \theta = \frac{\sqrt {19}}{10} [/tex]
Step-by-step explanation:
[tex] ( \sin \theta) ^{2} + ( \cos \theta) ^{2} = 1 \\ \\ \therefore \:( \sin \theta) ^{2} + { \bigg( \frac{{9} }{10} \bigg)}^{2}= 1 \\ \\( \sin \theta) ^{2} + \frac{81}{100}= 1 \\ \\ ( \sin \theta) ^{2} = 1 - \frac{81}{100} \\ \\ ( \sin \theta) ^{2} = \frac{100 - 81}{100} \\ \\ ( \sin \theta) ^{2} = \frac{19}{100} \\ \\ \sin \theta = \sqrt{\frac{19}{100} } \\ \\ \sin \theta = \frac{\sqrt {19}}{10} [/tex]
Simplify the product using the distributive property.
(4h + 6)(4h − 7)
16h2 − 52h + 42
16h2 + 4h − 42
16h2 + 52h + 42
16h2 − 4h − 42
Answers
Answer:
16h^2-4h-42
Step-by-step explanation:
4h times 4h is 16h^2
4h times -7 is -28h
4h times 6 is 24h
6 times -7 is -42
Combine like terms to get 16h^2-4h-42
the question is in the snapshot below!
Answers
Answer:
B
Step-by-step explanation:
opposite of ten is -10 and at least is grater that or equal to
Answer:
B is the correct answer
Step-by-step explanation:
I will give brainliest! What is the length of the segment that joins the points (-5, 4) and (6, -3)?
Answers
Answer:
Your answer would be √170, or square root of 170.
Which ordered pair is the solution to
the system of linear equations below?
(Lesson 8)
2x + y = -4
3x + 2y = -10
(0,4)
(6,-14)
(-3,2)
(2,-8)
Answers
Answer:
The fourth one! (2,-8).
BRAINLISTTTT
What is the simple interest on a $3,750 loan at a rate of 5% per year after 5 years?
a. $4,697.50
b. $13,125.00
c. $9,375.00
d. $937.50
Answers
Answer:
d.937.50
Step-by-step explanation:
After each year it adds 5% so 5x5 is 25%
25% of $3,750 is $937.50
hope this helps :)
Which function shown below has the greater rate of change? Explain. (Be sure to include the
rate of change for each in your explanation).
Answers
Answer:
Function 1 has a greater rate of change.
Step-by-step explanation:
Function 1 has a rate of change of 3 because 3x is the slope of the line.
Function 2 has a rate of change of 1.5, because when you use rise/run, you get a slope of 3/2.
Function 3 has a rate of change of 2. You use the formula (y2 - y1) / (x2 - x1) to find the slope. I used (0,0) and (1,2). When you plug in 2 for y2, 1 for x2, and 0 for y1 and 0 for x1, you get (2-0) / (1-0) which is 2/1. That equals a slope of 2.
Need explanation for this question please
Answers
Luther bought a juice box shaped like a rectangular prism. The juice box is 4 inches tall, 3 inches wide, and 1.5 inches deep. Each cubic inch of the juice box holds 0.55 fluid ounces of strawberry-kiwi punch.
How many fluid ounces of punch does the juice box hold?
Answers
Clint has a pizza that is cut into 4 equal slices. 3 pieces have pepperoni on them. The outher piece. only has cheese. what friction of the pizza only has cheese?
Answers
Answer:
1/4
Step-by-step explanation:
There are 4 slices
3 of the 4 slices has pepperoni
4-3=1
There is 1 slice without pepperoni, making 1/4th of the pizza pepperoniless.
Brainliest would be appriciated
Nathan buys a blender with a listed price of $65. Nathan uses a coupon that provides 20% off. What is the total final price Nathan pays?
Answers
Answer:
$52
Step-by-step explanation:
Given data
Price of blender= $65
Coupon = 20%
Let us find the amount of the coupon
=20/100*65
=0.2*65
=$13
Hence the amount of the coupon is $13
The total price Nathan pays is
=65-13
=$52
How does the number 32.4 change when you multiply it by 10 of the power of 2
Answers
If u=0, make t the subject of the following equation: s=ut+1/2at^2 Please look at image.
Answers
Answer:
[tex]t=\sqrt{\dfrac{2s}{a}}[/tex]
Step-by-step explanation:
The given equation is :
[tex]s=ut+\dfrac{1}{2}at^2[/tex]
Where
s is displacement
u is the initial velocity
a is acceleration
t is time
If u = 0, the above equation become,
[tex]s=\dfrac{1}{2}at^2\\\\t^2=\dfrac{2s}{a}\\\\t=\sqrt{\dfrac{2s}{a}}[/tex]
Hence, the value of t is equal to [tex]\sqrt{\dfrac{2s}{a}}[/tex].
12. You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of
the tree. You measure the angle of elevation from a point on the ground to the top of the tree
to be 59º. Estimate the height of the tree.
Answers
Answer:
tangent 59° = Tree height / 45 feet
Tree Height = 1.6643 * 45
Tree Height = 74.8935 Feet
Step-by-step explanation:
Solve the equation for x.
Answers
Answer:
-110
Step-by-step explanation:
[tex]-\frac{x}{10} =11[/tex]
-x=11×10=110
x=-110
what is the probability of rolling 15 with a pair of dice? answer in likely, unlikely, equally likely, or impossible
Answers
it is clearly impossible
It is impossible that the probability of rolling 15 with a pair of dice.
What is the probability?Probability refers to a possibility that deals with the occurrence of random events. The probability of all the events occurring need to be 1.
P(E) = Number of favorable outcomes / total number of outcomes
Required the probability of rolling 15 with a pair of dice.
But we know that in the roll of a pair of fair dice,
Then the sample space is as follows:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Hence, It is impossible that probability of rolling 15 with a pair of dice.
Learn more about probability here;
https://brainly.com/question/11234923
#SPJ2
Rona was driving on an asphalt road that had a 35 mile per hour speed limit posted. A lrr r deer jumped out from the side of the road causing Rona to slam on her brakes. Her tires left three skid marks of lengths 70 ft., 72ft., and71ft. The road had a drag factor of .78. Her brakes were operating at 95% efficiency. The police gave Rona a ticket for speeding. Rona insisted that she was driving under the speed limit. Who was correct (the police or Rona)
Answers
Answer:
The answer is "The cops were correct because the average speed would have been about 39.73 miles per hour as per the calculations."
Step-by-step explanation:
Finding the average skid distance:
[tex]Average=\frac{70+71+72}{3}=\frac{213}{3}=71[/tex]
Calculating the skid:
[tex]S=\sqrt{30 \times D \times f\times n}\\\\[/tex]
[tex]=\sqrt{30 \times 71 \times 0.78\times 0.95}\\\\=\sqrt{1,578.33}\\\\=39.73[/tex]
Solve for x.
I +27= -12
Enter your answer in the box
X
Answers
Answer:
you know that the coefficient of x is one so to find x the real equation is x+27= -12 so u cross over 27 to the right it will be -12-27=-39.
Which statements about the relationship between the two triangles below are true? Check all that apply.
Triangle A C D. Angle D is 60 degrees and angle A is 74.9 degrees. Triangle R S T. Angle R is 74.9 degrees and angle S is 45.1 degrees.
Angle C is congruent to angle T
Angle C is congruent to angle S
Angle D is congruent to angle T
Triangle D C A is congruent to triangle T S R
Triangle D C A is similar to triangle T S R
Triangle C A D is similar to triangle T R S
Triangle C A D is congruent to triangle T R S
Answers
Answer:
B C E on edge
Step-by-step explanation:
Answer:
correct answer BCE
Step-by-step explanation:
jack split 25 jelly beans between his brother and himself of that his brother has won more than twice the number jack has how many jellybeans does each have
Answers
Answer:
jack has 8 jellybeans and his brother has 17
Step-by-step explanation:
Jack have 8 jellies and his brother have 17 jellies.
What is fraction?A number expressed as a quotient, in which a numerator is divided by a denominator.
Total number of jellies= 25.
The jellies is to be divide between jack and his brother.
let us consider jack has 'x' jellies
Then, his brother have '2x+1'
So, x+ 2x+1=25
3x+1=25
3x=24
x=8
So, jack have 8 jellies and his brother have 2*8+1= 17 jellies.
Learn more about this concept here:
https://brainly.com/question/7022695
#SPJ2
The football team has a total of 50 jerseys. There are 20 medium-sized jerseys. What percent of the jerseys are medium-sized jerseys?
Answers
Answer:40%
Step-by-step explanation:20 divided by 50 is 0.4 then multiply that by 100 and get 40
Answer:
Well the equation for this would 50*.20 = ...
The first half of the answer would be 10.
BUT!
You need to subtract 10 from 50.
50-10 = 40!
There you go.
Step-by-step explanation:
2000 restaurants will grow about 3% every year, what is the equation that represents it. Y = restaurants and t = years
Answers
Answer:
Y= PV*(1 + g)^t
Step-by-step explanation:
Giving the following information:
Number of Restaurants (PV)= 2,000
Growth rate (g)= 3%
To calculate the number of restaurants after t years, we need to use the following formula:
Y= PV*(1 + g)^t
For example, for 10 years:
Y= 2,000*(1.03^10)
Y= 2,688
Add my Discord so we can be besties: BeepBoop #0214 12-14 only please!!
Answers
Answer: okay sir
Step-by-step explanation:
Which expression represents a rational number?
Answers
Answer:
C
Step-by-step explanation:
2/7+sqrt121 is just 2/7+11.
This is rational.
Answer:
2/7 + sqrt 121
Step-by-step explanation:
the square root of 121 is a whole number - 11, which is rational and makes the equation rational
HELP ME PLEASE
24d=12
What is d
-12c=12
What is c
Answers
A 20-pound bag of fertilizer costs $19.60. What is the cost
per pound?
Answers
Answer:
just do 19.60/20 = 0.98$/lbs
hope that answers your question
Step-by-step explanation: