I know you’re supposed to change the bounds and break up the integral, but for some reason, I can’t get the 44/3. Can someone explain how to solve this definite integral?
Answers
First, look for the zeroes of the integrand in the interval [0, 6] :
x² - 6x + 8 = (x - 4) (x - 2) = 0 ⇒ x = 2 and x = 4
Next, split up [0, 6] into sub-intervals starting at the zeroes we found. Then check the sign of x² - 6x + 8 for some test points in each sub-interval.
• For x in (0, 2), take x = 1. Then
x² - 6x + 8 = 1² - 6•1 + 8 = 3 > 0
so x² - 6x + 8 > 0 over this sub-interval.
• For x in (2, 4), take x = 3. Then
x² - 6x + 8 = 3² - 6•3 + 8 = -1 < 0
so x² - 6x + 8 < 0 over this sub-interval.
• For x in (4, 6), take x = 5. Then
x² - 6x + 8 = 5² - 6•5 + 8 = 3 > 0
so x² - 6x + 8 > 0 over this sub-interval.
Next, recall the definition of absolute value:
[tex]|x| = \begin{cases}x & \text{for }x \ge0 \\ -x & \text{for }x < 0\end{cases}[/tex]
Then from our previous analysis, this definition tells us that
[tex]|x^2 - 6x + 8| = \begin{cases}x^2 - 6x + 8 & \text{for }0<x<2 \text{ or } 4<x<6 \\ - (x^2-6x+8) & \text{for }2<x<4\end{cases}[/tex]
So, in the integral, we have
[tex]\displaystyle \int_0^6 |x^2-6x+8| \, dx = \left\{\int_0^2 - \int_2^4 + \int_4^6\right\} (x^2 - 6x + 8) \, dx[/tex]
Then
[tex]\displaystyle \int_0^2 (x^2 - 6x + 8) \, dx = \left(\frac13 x^3 - 3x^2 + 8x\right) \bigg|_0^2 = \frac{20}3 - 0 = \frac{20}3[/tex]
[tex]\displaystyle \int_2^4 (x^2 - 6x + 8) \, dx = \left(\frac13 x^3 - 3x^2 + 8x\right) \bigg|_2^4 = \frac{16}3 - \frac{20}3 = -\frac43[/tex]
[tex]\displaystyle \int_4^6 (x^2 - 6x + 8) \, dx = \left(\frac13 x^3 - 3x^2 + 8x\right) \bigg|_4^6 = 12 - \frac{16}3 = \frac{20}3[/tex]
and the overall integral would be
20/3 - (-4/3) + 20/3 = 44/3
Related Questions
The sales tax rate is 8% how much money is 8% of $14
Answers
Explanation:
12. After 80 years of 5.8% interest compounded monthly, an account has $102,393.44. What was the
original deposit amount?
Answers
Answer:
12195508,42
Step-by-step explanation:
A(1-R) NA is the depositR is the % presentN is the time period (exponent)A(1-5.8%)80=102,393.44An account has $102,393.44 after 80 years of 5.8% compounded monthly interest. The initial deposit was around $991.45.
What is meant by deposit?A deposit is money held at a bank. A deposit is a monetary transaction in which money is transferred to another party for safekeeping.A deposit is money placed in a bank account. Deposits are classified into two types: demand deposits and time deposits. Checking, savings, and money market accounts are all examples of demand deposit accounts. Certificates of deposit (CD) and individual retirement accounts are examples of time deposit accounts.Therefore,
Assume that each year has 52 weeks: then compounding occurs.
n = 52 × 80n
times and the rate for each compounding is
[tex]$i=\frac{5.8 \%}{52}=\frac{0.058}{52}$[/tex]
If the original deposit amount is
PV
then the future value
FV
is equal to
[tex]$$F V=P V \cdot(1+i)^n=P V \cdot\left(1+\frac{0.058}{52}\right)^{52 \cdot 80}$$[/tex]
We know the future value (which is given as $ 102,393.44), so calculating the present value is simple:
[tex]P V=F \frac{V}{(1+i)^n}=\frac{\$ 102,393.44}{\left(1+\frac{0.058}{52}\right)^{52 \cdot 80}}[/tex]
The simplest way to compute this is to use a calculator:
[tex]$P V \approx \$ 991.45 .$$[/tex]
We can simplify this expression with great precision just for fun. It is well known that
[tex]$\left(1+\frac{1}{n}\right)^n \rightarrow e$[/tex]
which easily implies that
[tex]$\left(1+\frac{a}{n}\right)^n \rightarrow e^a$[/tex]
Here,
[tex]$$\left(1+\frac{0.058}{52}\right)^{52 \cdot 80}=\left(\left(1+\frac{0.058}{52}\right)^{52}\right)^{80} \approx\left(e^{0.058}\right)^{80}=e^{0.058 .80}=e^{4.64}[/tex]
That is, [tex]P V \approx F V e^{-4.64}$.[/tex]
Hence, An account has $102,393.44 after 80 years of 5.8% compounded monthly interest. The initial deposit was around $991.45.
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The mass of a sheet of metal varies jointly with its area and its thickness. A sheet of metal of area 250cm2 and thickness 1mm has a mass of 200g. Find the formula which connects the mass M g, the area A cm2 and the thickness 1mm. Hence find the mass of a piece of metal of area 400 cm2 and thickness 3mm
Answers
The Formula of the relation is M = 8AT.
The mass of the metal sheet is 960 g.
The expression below shows the variation between the mass of the sheet (M), Area of the Sheet(A), and thickness (T).
Proportionality:
M ∝ ATRemoving the proportionality sign,
M = KATWhere:
K = constant of proportionality.make K the subject of the equation:
K = M/KT.................... Equation 1From the question,
Given:
M = 200 gA = 250 cm²T = 1 mm = 0.1 cmSusbtitute into equation 1
K = 200/(250×0.1)K = 8 g/cm³Formula:
M = 8AT................. equation 2Hence the formula of the relation is M = 8AT
If,
A = 400 cm²T = 3 mm = 0.3Substitute these values into equation 2
M = 8(400)(0.3)M = 960 gHence, the mass of the metal sheet is 960 g
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Raven can walk 2 2/5 miles in 3/5 hour. At this rate, how far can she walk in 1 hour? You might want to change 2 2/5 to an improper fraction in order to make the math easier for you to do.
Answers
The distance traveled by Raven in 1 hour at the given rate is 4.0 miles.
The given parameters:
Distance traveled by Raven = 2 2/5 milesTime of motion, t = 3/5 hourThe speed of Raven motion is calculated as follows;
[tex]V = \frac{Distance}{Time} \\\\V = Distance\ \times \ \frac{1}{Time} \\\\V = 2\frac{2}{5} \ miles \ \ \times \ \ \frac{1}{3/5 \ hr} \\\\V = \frac{12}{5} \ miles \ \ \times \ \ \frac{5}{3 hr} \\\\V = 4 \ mph[/tex]
The distance traveled by Raven in 1 hour at the given rate is calculated as follows;
[tex]D = 4 \times 1\\\\D = 4 \ miles[/tex]
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Answer:
1. given
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The impact on sampling of increasing the sample size is: There is no specific relationship between sample size and sampling error.
Answers
Using margin of error, it is found that increasing the sample size results in a smaller sampling error.
The equation for the margin of error is given by:
[tex]M = c\frac{s}{\sqrt{n}}[/tex]
In which:
c is the critical value according to the distribution used.s is the standard deviation, either of the sample or of the population.n is the sample size.From the equation, it can be noted that the margin of error is inversely proportional to the square root of the sample size, hence, increasing the sample size results in a smaller sampling error.
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A container can hold 2400 litres of water. (a) A large tap alone can fill the container in x hours. Write down an expression, in terms of r, for the amount of water that the large tap can dispense per minute. {b) A small tap alone will take 1 hour longer than the large tap to fill the co tainer. Write down an expression, in terms of x, for the amount of water that the small tap can dispense per minute. (c) When both taps are turned on at the same time, they can fill the container in 3 hours. Form an equation in x and shows that it reduces tox? -5x-3=0. X 20.
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Answer:
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Show work for brainliest!
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ANSWER:
y= -1/5x - 5
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Since it is passing through the point (-10, 3), substitute the numbers in for x and y which means: -3= -1/5*(-10) + b
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do the equation, and you should get the answer above!
Suppose you receive a shipment of 29 engines, two of which are defective. What is the probability of picking a sample of 4 engines with two bad engines in it? (Round approximations to the nearest thousandth.)
Answers
Answer:
0.4%
Step-by-step explanation:
2 are defective and 27 are good out of 29. Chance of picking 1 defective is 2/29 or .069. Chance of picking good is 27/29 or .931.
2 good and 2 bad is
.931 * .931 * .069 * .069 = .004
There is a 0.4 % chance of 2 bad.
Find the area of the equilateral triangle.
Answers
Answer:
try 6.93 for answer on equilqteral triangle
Carson bought 50 pounds of rice for his restaurant. He cooked the rice and wanted to evenly divide it into 9 containers to be served later. How many pounds of rice should Carson put in each container?
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Answer:
Step-by-step explanation: there you go
I thought of a number. This number double is 7 less than 29. What is my number? hurry please
Answers
Answer:
11
Step-by-step explanation:
11*2=22
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Answer:
11
Step-by-step explanation:
Let x be your number
Is means equals
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2x= 22
2x/2 = 22/2
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Answer:
The answer is A.
Step-by-step explanation:
1 plus 2x (276-5y) + 69= A.
The population of a slowly growing bacterial colony after t hours is given by
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Submit Question
Answers
A function is an expression that defines a relationship between an independent variable and a dependent variable.
The number of bacteria colonies after 2 hours is 220.
What is a function?It is an expression that defines a relationship between an independent variable and a dependent variable.
We have,
P(t) = 4t² + 27t + 150
Where t = time
P(t) = population of a slowly growing bacteria colony after a particular time
The growth rate after 2 hours:
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Divide 3/8 divided by 2/3 and express the answer in the simplest terms.
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Choose a way to solve.
Write or draw to explain.
2. Greg has Il shirts.
3 have long sleeves.
The rest have short sleeves.
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Answer:
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Step-by-step explanation:
Subtract the total amount of of shirts by the number of shirts with long sleeves to find your answer.
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Suppose that sin a= 15/17 and 90 degrees
Answers
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Answers
Answer:
62-0.75x8
Step-by-step explanation
$6=discount
8 tickets
Total=62.00
0.75 multiplied by 8= 6
62-6=56
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Solve the inequality below for z.
8z+3>3z-17
A.z>-4
B.z<-14/5
C.z<-4
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Answers
8z+3 > 3z-17
subtract 3 from both sides
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subtract 3z from both sides
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Hoped this helped.
[tex]BrainiacUser1357[/tex]
Solve the inequality 15 ≥ q + 3
Answers
Answer:
[tex]q\le \:12[/tex]
Step-by-step explanation:
Solving
[tex]\mathrm{Switch\:sides}[/tex] (Not a required step)
[tex]q+3\le \:15[/tex]
[tex]\mathrm{Subtract\:}3\mathrm{\:from\:both\:sides}[/tex]
[tex]q+3-3\le \:15-3\\=q\le \:12[/tex]
[tex]\bold{q\le \:12}[/tex]
Graphing
HURRY!! SUPER SUPER QUICK!!
i have 4,757 points!
ILL GIVE CROWN, STARS, and HEARTS if you answer correctly!
Problem:
6.02 Quiz: One-Step Multiplication or Division Inequalities
On k12
PLEASE ANSWER THEM ALL! VERY LEAST IS ALL YOU KNOW BUT PLEASE DO ALL, AMD CORRECTLY FOR CROWN!
Answers
Answer: m>-50
x<=3 3/4
at point -2.1 3rd arrow
at 1.33 4th arrow
at 4 2nd arrow
Step-by-step explanation:
The Hayes family and the Rodrigues family use their sprinklers last summer. The water output rate for the Hayes family sprinkler was 15 L per hour. The water Output rate for the Rodriguez family speakers is 30 L per hour.Families used their sprinklers for a combined total of 75 hour result in a total water of output of 1725 L how long was a sprinkler used?
Answers
Answer:
Hayes family: 35 hoursRodrigues family: 40 hoursStep-by-step explanation:
Let x and y represent the usage hours by the Hayes and Rodrigues families, respectively. The problem statement gives us relations that can be used to write equations for volume of use and for total hours.
15x +30y = 1725 . . . . . total liters of use
x + y = 75 . . . . . . . . . total hours of use
The second equation lets us write an expression for y:
y = 75 -x
This can be substituted into the first equation to give ...
15x +30(75 -x) = 1725
-15x + 2250 = 1725 . . . . . . simplify
-15x = -525 . . . . . . . . . . . subtract 2250
x = 35 . . . . . . . . . . . . . . divide by -15
y = 75 -35 = 40
The Hayes family used their sprinklers for 35 hours; the Rodrigues family used theirs for 40 hours.
Assume running consumes 95 cal per mile. If you run 9- minute miles, what is your average power output in watts
Answers
95/9= about 10.5 cal/min
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(5x + 3) + (9x + 5)
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Which triangle is a 30°-60°-90° triangle?
Answers
Answer:
-90
Step-by-step explanation:
it's right
Answer:
The first one.
Step-by-step explanation:
I took the quiz and got a 100, hope this helps :))
The amount of money charged to borrow money, expressed as a percentage,
is called the
A. collateral
B. interest rate
O C. principal
D. loan term
Answers
Answer:
Interest rate
Step-by-step explanation:
The amount owed is called the principal and the price of borrowing money is called interest. Some people spend a day's pay (or more) per week repaying the interest and principal owed on car loans, credit card bills, student loans, and other consumer debts.
A projectile is thrown upward so that its distance above the ground after t seconds is h = -156 + 510t. What is its maximum height?
Answers
For the parabola:
h(t) = -at2 + bt, the maximum occurs at -b/2a
For h(t) = -16t2 + 576t that is -576/(-2)(-16), or -576/32, or 18.
The projectile is at its highest point at 18 seconds.
A2 + B2
Best answer gets a brainliest and thank and 5 star
Answers
Answer:
a+b)(a-b)
(Step-by-step explanation:
Translate this sentence into an equation.
46 is the sum of 20 and Han's savings.
Use the variable h to represent Han's savings. I
Answers
Answer:
20+h = 46
Step-by-step explanation:
it doesn't ask to solve it so that's the answer
Answer :-
[tex]20 + h = 46[/tex]• 46 is the sum of 20 and Han's savings. .
So here Question tell that 46 is the sum of Han's Saving .Then it says that to write the equation in this sentence .
• Use the variable h to represent Han's savings.
So 20 and h = 20 + h = 46 .Hope this helps you !!