Find the area of the region enclosed by the graphs of these equations. (CALCULUS HELP)
Answers
Answer:
[tex]\displaystyle A = \frac{20\sqrt{15}}{3}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightEquality Properties
Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of EqualityAlgebra I
Terms/CoefficientsGraphingExponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Calculus
Derivatives
Derivative Notation
Derivative of a constant is 0
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Area - Integrals
U-Substitution
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
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
F: y = √(15 - x)
G: y = √(15 - 3x)
H: y = 0
Step 2: Find Bounds of Integration
Solve each equation for the x-value for our bounds of integration.
F
Set y = 0: 0 = √(15 - x)[Equality Property] Square both sides: 0 = 15 - x[Subtraction Property of Equality] Isolate x term: -x = -15[Division Property of Equality] Isolate x: x = 15G
Set y = 0: 0 = √(15 - 3x)[Equality Property] Square both sides: 0 = 15 - 3x[Subtraction Property of Equality] Isolate x term: -3x = -15[Division Property of Equality] Isolate x: x = 5This tells us that our bounds of integration for function F is from 0 to 15 and our bounds of integration for function G is 0 to 5.
We see that we need to subtract function G from function F to get our area of the region (See attachment graph for visual).
Step 3: Find Area of Region
Integration Part 1
Rewrite Area of Region Formula [Integration Property - Subtraction]: [tex]\displaystyle A = \int\limits^b_a {f(x)} \, dx - \int\limits^d_c {g(x)} \, dx[/tex][Integral] Substitute in variables and limits [Area of Region Formula]: [tex]\displaystyle A = \int\limits^{15}_0 {\sqrt{15 - x}} \, dx - \int\limits^5_0 {\sqrt{15 - 3x}} \, dx[/tex][Area] [Integral] Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle A = \int\limits^{15}_0 {(15 - x)^{\frac{1}{2}}} \, dx - \int\limits^5_0 {(15 - 3x)^{\frac{1}{2}}} \, dx[/tex]Step 4: Identify Variables
Set variables for u-substitution for both integrals.
Integral 1:
u = 15 - x
du = -dx
Integral 2:
z = 15 - 3x
dz = -3dx
Step 5: Find Area of Region
Integration Part 2
[Area] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle A = -\int\limits^{15}_0 {-(15 - x)^{\frac{1}{2}}} \, dx + \frac{1}{3}\int\limits^5_0 {-3(15 - 3x)^{\frac{1}{2}}} \, dx[/tex][Area] U-Substitution: [tex]\displaystyle A = -\int\limits^0_{15} {u^{\frac{1}{2}}} \, du + \frac{1}{3}\int\limits^0_{15} {z^{\frac{1}{2}}} \, dz[/tex][Area] Reverse Power Rule: [tex]\displaystyle A = -(\frac{2u^{\frac{3}{2}}}{3}) \bigg|\limit^0_{15} + \frac{1}{3}(\frac{2z^{\frac{3}{2}}}{3}) \bigg|\limit^0_{15}[/tex][Area] Evaluate [Integration Rule - FTC 1]: [tex]\displaystyle A = -(-10\sqrt{15}) + \frac{1}{3}(-10\sqrt{15})[/tex][Area] Multiply: [tex]\displaystyle A = 10\sqrt{15} + \frac{-10\sqrt{15}}{3}[/tex][Area] Add: [tex]\displaystyle A = \frac{20\sqrt{15}}{3}[/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
Extra Points and brainliest! Question below
Answers
Answer:
14 one fourths
Step-by-step explanation:
First, we convert 7/2 into a fraction with the denominator of a four. To do this, we multiply the top and bottom by 2. 7*2 is 14, and 2*2 is 4, which gives us 14/4. This means that there are 14 one fourths in 7/2. Hope this helps!
Answer:
14 fourths
Step-by-step explanation:
Hey There!
So to solve this problem we need to divide 7/2 by 1/4
[tex]\frac{7}{2}divided by\frac{1}{4} =14[/tex]
Therefore 14 fourths can go into 7/2
Which inequality is true if p=3?
A. 3p <9
B. 13 < 4p
C. 5p > 15
D.8 > 2p
Answers
Answer:
D
Step-by-step explanation:
2×3 = 6 which is less than 8
Elis, Raine and Avery buy a boat for $33 000.
Elise pays $12 000, Raine pays $6000 and Avery pays the rest.
Six years later they sell the boat for $25 300.
They share the money in the same ration that they bought the boat.
How much money does Avery lose on the sale of the boat?
Answers
Amount Avery paid: 32,000-12,000-6,000 = 15,000
Divide portion she paid by total cost:
15,000/33,000
Simplify to 15/33
Multiply the amount the boat was sold for by her portion:
25,300 x 15/33 = 11,500
Subtract the amount she got from the amount she paid:
15,000 - 11,500 = 3,500
She lost $3,500
Answer:
yes
Step-by-step explanation:
yes!
PLEASE, PLEASE HELP WITH THIS SOON!
A restaurant customer left $1.10 as a tip. The tax was 4% and the tip was 10% of the after tax cost. Which information is not needed to compute the bill after tax and tip? What was the total bill?
Answers
Answer:
all info is needed answer of the total bill is:275 us
Step-by-step explanation:
10x1.10=11= tax
100/z=4/11
275
--Please help, ill give 30 points and brainiest if your correct!!--
Lisa and Alice are playing a game. Each player either receives or has to pay play money based on the results of their spin. The table lists how much a player receives or pays for various spins. What color has the greatest amount?
Red
Pays $5
Blue
Receive $4
Yellow
Pay $1
Green
Receive $3
Orange
Pay $2
Answers
Answer:
The amount in table as positive negative numbers: -5, 4, -1, 3, -2.
The spinner lands on red the change to Lisa's amount of money is -$5.
Step-by-step explanation:
Consider the provided table.
Part(A)
The table list how much a player receives or pays for various.
Paying some money is represented by a negative sign while receiving money is represented by a positive sign.
For red they needs to pay $5 which can be written as -5.
For blue they received $4 which can be written as 4.
For yellow they needs to pay $1 which can be written as -1.
For green they received $3 which can be written as 3.
For orange they needs to pay $2 which can be written as -2.
So, the amount in table as positive negative numbers: -5, 4, -1, 3, -2.
Part(B)
Now, we need to describe the change to Lisa’s amount of money when the spinner lands on red.
If spinner lands on red then she needs to pay $5.
Thus, if the spinner lands on red the change to Lisa's amount of money is -$5.
A square notepad has sides that are 14 centimeters long. What is the notepads perimeter?
Answers
14x4=56
Answer:
56 cm (22.4 in)
Step-by-step explanation:
14+14+14+14=56 or 14x4=56
how do i solve this and what is the answer?
Answers
Answer:
I totally get why you are asking , how do I solve this. there is so much implied about this
Step-by-step explanation:
1st you have to recognize that ∠ACB ( in my drawing ) has that same angle as the arc from A to B , so that's implied. next you have to recognize that the triangle formed by ACB with the 124 degree angle at C is an isosceles and therefore has identical angles at A and B so that's 180 = 124 + 2A , so A = 28° and so does B , then , b/c we know that small angle... you also have to know, that the line AC is perpendicular to AD. that they from a right angle and there for... you now know , that ∠BAD is the other part of that 90 or 90= 28 + ∠BAD so ∠BAD = 62°
I went through that kinda fast, there was several "leaps" of using logic to get to the correct answer , using things you know from prior understanding about angles , circles and geometry, you are putting all of the parts together here. :/ kinda daunting in some ways.
HELP ASAP PLS!!
what is composite area?? How do we use it in the real world???
Answers
Answer:
A composite figure (or shape) is one that can be separated into more than one of the basic figures. Examples of using it in the real world is building cars or houses.
What is 21+G as an algebraic expression?
Answers
Answer:
i dont know
Step-by-step explanation:
lol
Answer:
(20 + 1) + G
Step-by-step explanation:
Using the figure below, what is the trigonometric ratio of cos C?
Answers
Answer:
8/9 (choice b)
Step-by-step explanation:
The trigonometric ratios are based on the ratios of different sides.
If you remember SOH CAH TOA =
sine (sin) opposite hypotenuse
cosine (cos) adjacent hypotenuse
tangent (tan) opposite adjacent
__________________________
This is a nuemonic for these ratios.
the opposite side is the side that is directly across from the reference angle, the hypotenuse is the longest side, and the adjacent side is the side other than these two.
So cos C = adjacent side / hypotenuse side.
Since 18 is greater than 16 and 9, 18 is the hypotenuse.
the adjacent side is the side other than the opposite side and the hypotenuse which is 16
therefore cos C = 16 / 18 = 8 / 9.
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Answers
Answer:
1
Step-by-step explanation:
True or false - The only way to charge an object with static electricity is by rubbing it against another object.
Answers
Answer:
This is false
Step-by-step explanation:
A town has a population of 7000 and grows at 4% every year. What will be the population after 13 years, to the nearest whole number?
Answers
Answer:
Step-by-step explanation:
7,000 × 1.04¹³ = 11,656
Which is the equation of a line that has a slope of -7 and a Y intercept of 2
Answers
Answer:
y = -7x + 2
Step-by-step explanation:
you have $63.50. you earn additional money by raking leaves. you purchase a sweatshirt $44.99 and you have $53.51 left. how much money did you earn raking leafs?
Answers
5.5 + n = 10 please help
Answers
Answer:
n = 4.5
Step-by-step explanation:
10 - 5.5 = n
Answer:
I feel like its 4.5
Step-by-step explanation:
Factor the follow expression:
8x−10
Select one:
2(x−5)
2(4x−5)
8(x−10)
2(4x−10)
Answers
Answer:
2(4x-5)
Step-by-step explanation:
find the greatest common factor (biggest number that goes into 8 and 10)
it is two
divide both by two
put 2 on the outside to symbolize that you multiply the inside by 2
2(4x-5)
Answer:
[tex]2(4x - 5)[/tex]
Step-by-step explanation:
1) Find the Greatest Common Factor (GCF).
[tex]gcf = 2[/tex]
2) Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
[tex]2( \frac{8x}{2} - \frac{10}{2} )[/tex]
3) Simplify each term in parentheses.
[tex]2(4x - 5)[/tex]
Therefor, the answer is Option B.
Patrick is reading a book about the history of baseball. The book is 210 pages long; 140 of the pages have illustrations on them. If Patrick opens the book to a random page, what is the probability that the page will have an illustration on it?
Answers
Answer:
Probability: 140/210= 14/21Round to the nearest hundredth 94.855
Answers
Your welcome :)
Evaluate the expression if a = -2, b= 7. and c= -3
ac + b=
Answers
Answer:
13
Step-by-step explanation:
plug in the numbers
-2(-3) + 7 = ?
6 + 7 = ?
13
Hope this helped! Have a nice day! Plz mark as brainliest!!!
-XxDeathshotxX
The sum of 3 times the value of x and 2 is equal to less than 5 times the value of x. Which equation can be used to find the value of x?
A. 3x+2=4-5
B. 2x+3=4-5x
C. 3x+2=5x-4
D. None of these
Answers
sum is the answer to addition problem.
less than is subtraction and "times" is multiplication.
So we have
3x+2=5x-4
So the answer is C.
Also, i'm bored so imma solve it.
3x+2=5x-4
-2 -2
3x=5x-6
-5x -5x
-2x=-6
/-2 /-2
x=3
---
hope it helps
6 ft = ____ yd
6 ft = 6 x 1 ft
= 6 x ____ yd
= ____ yd
Answers
Answer:
6 ft = 2 yd
6 ft = 6 x 1 ft
= 6 x 2 yd
= 12 yd
Which conditions are necessary for hail to form?
Answers
Answer:
Normally in intense thunderstorm clouds. The temperature should be below 32 degrees Fahrenheit in the clouds where the hail forms, typically there is also large water droplets and strong updrafts.
Answer: The freezing temperature of the air below the cloud causes the water droplets to freeze.
Step-by-step explanation: Weather conditions must be just right in order for hail to occur. The most important factors are dark,gray clouds alongside thunder and lightning in freezing or below freezing temperatures. When thunder is strong enough to raise water droplets up into freezing temperatures,hail is formed.
ANSWER ONE OF THESE FAST PLEASE!!!
Answers
Answer:
x7
Step-by-step explanation:
what is the volume if this rectangular prism
Answers
Answer:
24.2^3cm is the correct answer .
answer pls>>>>>.
pls
Answers
I used completing the square method were you draw a square and multiple everything across
Hope it helps!!
Have a nice day or night :O
Use the comment feature for any additional questions you have
A sample of radium has a weight of 1.5 mg and a half-life of approximately 6 years.
How much of the sample will remain after 1 year?
Answers
Answer:
Step-by-step explanation:
1.5×0.5^(⅙) ≅ 1.3 mg
(08.01)
Two friends went to get ice cream sundaes. They each chose a flavor of ice cream from a list of vanilla and chocolate and toppings from a list of hot fudge,
strawberries, sprinkles, peanuts, and whipped cream. Use the sets below describing their choices and find B'
Let A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
Let B = {vanilla, hot fudge, sprinkles, whipped cream}
Let C = {chocolate, hot fudge, peanuts, whipped cream}
Answers
Answer:
A
Step-by-step explanation:
njsusnhshbwhjkahbkans
helpppppp ill give brainliest answer if you know
Answers
Answer: 522.67 mm^3
Step-by-step explanation:
use formula
10. Ayden travels from Orlando to Tampa to see his grandparents. If the trip is 897,600 feet round trip, how many miles does Ayden travel one way? A 85 miles B 170 miles C 255 miles D 340 miles 11. Jason bought 4.5 kilograms of cream me of cream?
Answers
Bell Lah-Lah she will do