Evaluate the line integral, where c is the given curve. (x + 9y) dx + x2 dy, c c consists of line segments from (0, 0) to (9, 1) and from (9, 1) to (10, 0)

Answers

Answer 1

[tex]\displaystyle\int_C(x+9y)\,\mathrm dx+x^2\,\mathrm dy=\int_C\langle x+9y,x^2\rangle\cdot\underbrace{\langle\mathrm dx,\mathrm dy\rangle}_{\mathrm d\mathbf r}[/tex]

The first line segment can be parameterized by [tex]\mathbf r_1(t)=\langle0,0\rangle(1-t)+\langle9,1\rangle t=\langle9t,t\rangle[/tex] with [tex]0\le t\le1[/tex]. Denote this first segment by [tex]C_1[/tex]. Then

[tex]\displaystyle\int_{C_1}\langle x+9y,x^2\rangle\cdot\mathbf dr_1=\int_{t=0}^{t=1}\langle9t+9t,81t^2\rangle\cdot\langle9,1\rangle\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(162t+81t^2)\,\mathrm dt[/tex]
[tex]=108[/tex]

The second line segment ([tex]C_2[/tex]) can be described by [tex]\mathbf r_2(t)=\langle9,1\rangle(1-t)+\langle10,0\rangle t=\langle9+t,1-t\rangle[/tex], again with [tex]0\le t\le1[/tex]. Then

[tex]\displaystyle\int_{C_2}\langle x+9y,x^2\rangle\cdot\mathrm d\mathbf r_2=\int_{t=0}^{t=1}\langle9+t+9-9t,(9+t)^2\rangle\cdot\langle1,-1\rangle\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(18-8t-(9+t)^2)\,\mathrm dt[/tex]
[tex]=-\dfrac{229}3[/tex]

Finally,

[tex]\displaystyle\int_C(x+9y)\,\mathrm dx+x^2\,\mathrm dy=108-\dfrac{229}3=\dfrac{95}3[/tex]

Answer 2

Final answer:

To evaluate the given line integral, you need to parameterize the curve composed of two segments. Then, you solve the integrals along the two separate segments and take the sum.

Explanation:

To evaluate the line integral where the curve c consists of line segments from (0, 0) to (9, 1) and from (9, 1) to (10, 0) for the function (x + 9y) dx + x² dy, we need to parameterize the curve and solve the integral.

First, we parametrize the line segments. For the segment from (0, 0) to (9, 1), we have x = 9t and y = t for t in [0, 1]. For the segment from (9, 1) to (10, 0) let x = 9 + s and y = 1 - s for s in [0, 1].

Then, we integrate along the first and second segments separately with respect to t. For the first segment, the integral is ∫(9t + 9t) d(9t) + ∫(81t²) dt = ∫18t dt + ∫81t² dt. For the second segment, the integral is ∫(9 + s + 9(1 - s)) ds + ∫(81 + 2 * 9 * s + s²) d(1 - s) = ∫ds + ∫s² ds.

We then evaluate these integrals over the respective bounds, sum them up and that gives the line integral along the path described.

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Related Questions

Expanded form for 4082

Answers

4000+80+2
your so welcome.

Find two numbers with a sum of 20 and a difference of 14.

Answers

17 and 3 is your answer

17 + 3 = 20

17 - 3 = 14

hope this helps

Greg drank 5 cups of water on Monday. He drank 4 cups of water on Tuesday. How many 1/4 cup servings of water did Greg drink on Monday and Tuesday combined?

Answers

Start the problem by first finding out how many 1/4 cups of water equal 1 full cup of water. [tex]\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1[/tex], therefore we can see that every 1 cup of water is equivalent to 4 1/4 cups of water. Now we will find the total number of cups of water Greg drank on monday and tuesday by adding 5 + 4 = 9. Multiply this number by 4 and you have the total number of 1/4 cups of water Greg drank on Monday and Tuesday combined.

So what I did was take 5 and 4 and then add them which would be 9, after that take 1/4 and make 9 one/forth. So it would look like 1/4+ 1/4+ 1/4+ 1/4+ 1/4 + 1/4+ 1/4+ 1/4+ 1/4= 2.25

You are planning a party and want to know how many cans of soda to buy. a genie offers to tell you either the mean number of cans guests will drink

Answers

Is there more to this question??

Find the cost per pound of a "house blend" of coffee that is made from 20 lb of Central American coffee that costs $7 per pound and 30 lb of South American coffee that costs $4.50 per pound

Answers

Well first things first, you want to multiply 20 and 7, this way, you get $140 per lb. After this, you want to multiply 30 by 4.50, and you get $135 per lb. You want to add these two numbers, 140 + 135, and get the total sum of $275. And to find the per pound of this sum, the "house blend" you simply will need to add 7 and 4.50, to get $11.50 per pound.

The point(3,7) is reflected in the y axis what are the coordinates now

Answers

When a point P(a, b) is reflected about the y-axis, the coordinates of the reflected point are P'(-a, b).

Thus, the reflection of point (3, 7) is (-3, 7), as shown in the picture.

Answer: (-3, 7)

For a fixed loan amount, what happens to the monthly payment as the annual rate increases? a. as the annual rate increases, the monthly payment increases c. as the annual rate increases, the monthly payment stays the same b. as the annual rate increases, the monthly payment decreases d. no correlation

Answers

If the annual rate increases nothing will happen to the monthly payment, so the answer is C as the annual rate increases, the monthly payment stays the same. Once the fixed-rate payment (loan) is processed there is no chance for any changes as this procedure means that lender is under a fixed-rate loan each payment period.

1. 3/5x -7 = 23

A. x = 40
B. x = 45
C. x = 50
D. x = 55

2.When Patrick and Cameron went bowling at Victory Lanes, they spent a total of $42. This included rental of shoes at $3.50 per pair. Patrick bowled 2 games, and Cameron bowled 3 games.
How much did each game cost?
A. $6.00
B. $7.00
C. $7.70
D. $8.75
3.A local movie theater offers a special deal for birthday parties. The price, which covers the cost of the tickets and a box lunch, includes a flat fee of $75 plus $5 per ticket. Sarah’s parents spent $150 altogether at the theater to celebrate her birthday.

Which equation can be used to determine how many children, x, were in the group?
A. 75/150=5x
B. 150/75=5x
C. 150 = 5x – 75
D. 150 = 5x + 75

4. 35 + 0.25x = 53.75
A. x = 75
B. x = 140
C. x = 215
D. x = 250

5.T he perimeter of this isosceles triangle is 47 cm.
top point. (B) left bottom point (A) right bottom point (C) in between 12cm
What is the length of each of its legs (it doesn't have legs)?
A. 15 cm
B. 17.5 cm
C. 19.5 cm
D. 35 cm

Answers

3/5x - 7 = 23
3/5x = 23 + 7
3/5x = 30
x = 30/(3/5)
x = 30 * 5/3
x = 150/3
x = 50 <==
=====================
3.50(2) + 5g = 42
7 + 5g = 42
5g = 42 - 7
5g = 35
g = 35/5
g = 7 <== each game costs $ 7
===================
150 = 5x + 75
===================
35 + 0.25x = 53.75
0.25x = 53.75 - 35
0.25x = 18.75
x = 18.75 / 0.25
x = 75 <===
===================
if one leg is 12.....and an isosceles triangle has 2 equal sides....(47 - 12) / 2 = 35/2 = 17.5 <==

Bob is a high school basketball player. his probability of making a free throw is 0.70. let x denote the number of hits in 10 shots. . find the probability that bob makes at least 9 hits. the probability is _____________.

Answers

Xjxjjxjxbdbdndbdbbdndhdhdhxhdhdhhdhdhdhdbdhdhdhdhhd

To extend the Sieve of Eratosthenes to 200, what is the largest prime whose multiples would have to be considered?

Answers

The largest prime that needs to be checked for factors of a number n is floor(sqrt(n)), or |sqrt(n)| in some notations, meaning the largest integer that does not exceed sqrt(n).

For example, if checking 169, whose squre-root is 13. There will be NO prime factors below 13 (1 is not a prime). So check for all prime factors up to and including the largest prime equal to or less than sqrt(n).

Therefore, the largest prime whose multiples need to be considered to extend the Sieve of Eratosthenes to 200 is 13.

The Sieve of Eratosthenes is a method used to find all prime numbers up to a given limit by iteratively marking the multiples of each prime number. To extend the Sieve of Eratosthenes to 200, determine the largest prime whose multiples need to be considered.

Begin by creating a list of numbers from 2 to 200. Mark 2 as the first prime number. Proceed to eliminate all multiples of 2 from the list (4, 6, 8, ...).

Move to the next unmarked number, which is 3. Mark it as a prime and eliminate all its multiples from the list (6, 9, 12, ...).

Continue this process for each unmarked number: marking it as a prime and eliminating its multiples. The next unmarked number after 3 is 5.

Continue this process until you reach the square root of the upper limit, which is √200 ≈ 14.14. Therefore, the largest prime whose multiples need to be considered to extend the Sieve of Eratosthenes to 200 is 13. Multiples of primes greater than 13 that fall below 200 have already been eliminated during the sieving process.

Two negative integers are five minutes apart on the number line, and their product is 126. What is the sum of the two integers?

Answers

First I will assume that this problem can be reworded as follows: "Two negative integers are five units apart on the number line."

Let the first (the smaller) negative integer be x and the second y. Note that y, being bigger than x, can be represented by y=x+5.

First find the 2 integers. Their product is 126, so x(x+5)=126.
Expanding, x^2 + 5x - 126 = 0. Using the quadratic formula to solve this,

-5 plus or minus sqrt(529) -5 plus or minus 23
x = --------------------------------------- = -----------------------------
2 2
-5 -23
We are interested only in negative integers, so take x = ------------
2

Then x = -28/2, or x = -14. Then y is x+5, or -14+5 = -9.

Check: does the product of -9 and -14 equal 126, as it must? YES.

which pair of fractions is equivalent to this pair : 3/5 and 2/7

a. 21/35 & 10/35
b. 10/35 & 7/35
c. 7/35. & 5/35
d. 24/35 & 12/35

Answers

I believe the answer would be A.
Hope this helps!

First you want to get a common denominator. In this case it is 35. So multiply whatever number works to get them equal to 35.

3/5(7/7) = 21/35
2/7(5/5) = 10/35

A.21/35 and 10/35

Hope this helps!

Each side of a square is increasing at a rate of 5 cm/s. at what rate is the area of the square increasing when the area of the square is 49 cm2? 70 cm2/s

Answers

This problem is solved using the chain rule. the area of the square is f(t) and the length of the side is g(t) f(t)=g(t)^2 g'(t)=5 Using the chain rule f'(t)=2*g(t)*g'(t) The value of g(t) is sqrt(49) which is 7. g'(t) is given as 5 cm/s f'(t)=2*7*5=14*5=70cm^2/s

At full power, how long would it take for the car to accelerate from 0 to 64.0 mph ? neglect friction and air resistance. express your answer in seconds.

Answers

The amount of time it would take for a car to accelerate from 0 to 64.0 mph can vary based on the vehicle. Basic vehicles generally can accelerate in under 6 seconds to 64 mph, while exotic cars can reach that speed somewhere in between 3 and 4 seconds.

Earth is about 4.5 billion years old and written human history extends back about 10,000 years. Suppose the entire history of Earth is represented with a 1000-meter-long timeline, with the birth of Earth on one end and today at the other end.
2 billion years is represented by how many meters of the timeline?

Answers

444.44 meters.

Hope that helps

Last year, Amanda had
$10,000
to invest. She invested some of it in an account that paid
6%
simple interest per year, and she invested the rest in an account that paid
10%
simple interest per year. After one year, she received a total of
$760
in interest. How much did she invest in each account?

Answers

Let $x be invested in the account that paid 5% simple interest.
Let $y be invested in the account that paid 10% simple interest.

The total invested is $10,000, therefore
x + y = 10000            (1)

The interest amounts earned are respectively
x*1*0.06 = 0.06x
y*1*0.10 = 0.10y

The total interest earned is $760, therefore
0.06x + 0.10y = 760
Multiply through by 10 to obtain
0.6x + y = 7600              (2)

Subtract (2) from (1).
x + y - (0.6x + y) = 10000 - 7600
0.4x = 2400
x = $6,000
From )1), obtain
y = 10000 - 6000 = $4,000

Answer:
$6,000 in the account paying 6% simple interest, and
$4,000 in the account paying 10% simple interest.

Suppose you buy a piece of office equipment for $9,000.00. After 5 years you sell it for a scrap value of $2,000.00. The equipment is depreciated linearly over 5 years. The value of the piece of equipment after 2 years is (rounded to the nearest whole dollar)

Answers

Residual value = $2000

$7000/5 = $1400

Yearly depreciation = $1400

So in 2 years total depreciation = $2800.

Value = Purchase Price - Total Depreciation
Value = $9000 - $2800
Value after 2 years = $6200

The value of the piece of equipment after 2 years is $6,200.00.

What is a Linear equation?

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. Sometimes, the aforementioned is referred to as a "linear equation of two variables," with y and x serving as the variables.

The equipment is depreciated linearly, which means that its value decreases at a constant rate each year. We can find the rate of depreciation by subtracting the scrap value from the purchase price, and then dividing by the number of years of useful life:

Rate of depreciation = (Purchase price - Scrap value) / Useful life

Rate of depreciation = (9000 - 2000) / 5

Rate of depreciation = 1400

This means that the equipment loses $1,400 of value each year. After 2 years, the value of the equipment will be:

Value after 2 years = Purchase price - (Rate of depreciation x Number of years)

Value after 2 years = 9000 - (1400 x 2)

Value after 2 years = 6200

Therefore, the value of the piece of equipment after 2 years is $6,200.00.

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Suppose that f(t)f(t) is continuous and twice-differentiable for t≥0t≥0. further suppose f′′(t)≤3f″(t)≤3 for all t≥0t≥0 and f(0)=f′(0)=0f(0)=f′(0)=0. using the racetrack principle, what linear function g(t)g(t) can we prove is greater than or equal to f′(t)f′(t) (for t≥0t≥0)?

Answers

(x) is continuous and twice differentiable for ±≥ 0. Suppose f”( ±) ≤ 5 for all ±≥ 0 and f (0) = f’(0) = 0. The linear function with a derivation of 0 is y = 0. Thus we have g(t) = 0. The quadratic function with concavity 5 and initial slope 0 is y = 5/2 x^2 + 0x. Therefore, h(t) = (5/2)x^2

Final answer:

Given the conditions, the function g(t)=3t is always greater than or equal to f'(t), as it represents the maximum possible velocity considering the maximum constant acceleration of 3 over time.

Explanation:

Given that f(t) is a continuous and twice-differentiable function for t≥0 and that f''(t)≤3 for all t≥0, the function f(t) represents the position of an object over time, with f'(t) and f''(t) being the velocity and acceleration, respectively. Further, given that f(0)=f'(0)=0, it is stated that at time t=0, the object is at the origin and is at rest.

Using the Racetrack Principle, we know that velocity must be less than or equal to the maximum acceleration times time. In plain terms, the actual speed of a body at a time 't' (f'(t)) could not be more than the distance it would have covered if it was accelerating at the maximum possible rate from rest. Since f''(t)≤3, the maximum possible acceleration is 3.

Thus, for all t≥0, if we consider the linear function g(t)=3t, we can say g(t) is greater than or equal to f'(t). Because even if the acceleration was at the maximum value of 3 for the entire duration, the velocity wouldn't exceed 3t.

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Kate has a serving account that contains $230. She decides to deposit $5 each month from her monthly earnings for baby-sitting after school. Write an expression to find how much money Mata will have in her savings account after x months. Let x represent the number of months. Then find out how much she will have in her account after 1 year

Answers

She will have $290 after 1 year

Choose the fraction that goes in the blank. six over ten >_______> one over two A one over two B one over four C two over three D three over four

Answers

Final answer:

To determine which fraction is greater, we convert both fractions to decimal form. Six over ten is greater than one over two.

Explanation:

To determine which fraction is greater, six over ten or one over two, we can convert both fractions to decimal form and compare them. Six over ten can be simplified to three over five, which is equal to 0.6. One over two is equal to 0.5. Since 0.6 is greater than 0.5, we can conclude that six over ten is greater than one over two.

The price of 6 pretzels is $5.10. Simon and Sofia bought 8 pretzels and shared the cost equally. How much did each person pay?

Answers

$4.68 because each pretzel is $1.17 so 8 pretzels are 9.36 ÷ by 2 = $4.68

Sarah's craft project uses pieces of yarn that are 1/8 long. She has a piece of yarn that is 3 yards long. How many 1/8 yard pieces can she cut and still have 1 1/4 yards left?

Answers

well, first off let's check how much is 3 minus the 1 1/4, so the leftover we check how many 1/8 go into it.

[tex]\bf 1\frac{1}{4}\implies \cfrac{1\cdot 4+1}{4}\implies \cfrac{5}{4}\\\\ -------------------------------\\\\ 3-1\frac{1}{4}\implies 3-\cfrac{5}{4}\impliedby \textit{LCD of 4}\implies \cfrac{(4\cdot 3)-5}{4}\implies \cfrac{7}{4} \\\\\\ \textit{now, how many times does }\frac{1}{8}\textit{ go into }\frac{7}{4}?\implies \cfrac{\quad \frac{7}{4}\quad }{\frac{1}{8}}\implies \cfrac{7}{4}\cdot \cfrac{8}{1} \\\\\\ \cfrac{56}{4}\implies 14[/tex]

-3.1 * (- 2.9) Please help! Math.

Answers

8.99 would be your answer

The answer is

(-3.1)(-2.9)

=8.99

Is 25/99 a terminating or repeating decimal

Answers

25/99 = 0.2525252525

Do you see that 25 repeats?

Repeating decimal for sure.

The graph of y = –3x + 4 is:
A. a point that shows the y-intercept.
B. a line that shows the set of all solutions to the equation.
C. a point that shows one solution to the equation.
D. a line that shows only one solution to the equation.

Answers

Answer: The answer is B, just got this one on a test.

Step-by-step explanation:

x+2y=8 in slope intercept form

Answers

This question is super easy, all you need to do is isolate the y variable. When you want to isolate it, you have to get rid of other numbers by doing the opposite sign.
x+2y=8 Subtract the x.
2y=8-x Divide by two.
y=4-1/2x

[tex]x+2y=8[/tex] can be written in a slope-intercept form as [tex]y = 4-\dfrac{x}{2}[/tex].

Slope

A slope is also known as the gradient of a line is a number that helps to know both the direction and the steepness of the line.

Given to us,

[tex]x+2y=8[/tex]

To find the slope-intercept form

To find the function, simply isolate y.

[tex]x+2y=8\\2y = 8-x\\y = \dfrac{8-x}{2}\\\\y = \dfrac{8}{2}-\dfrac{x}{2}\\\\y = 4-\dfrac{x}{2}[/tex]

Hence, [tex]x+2y=8[/tex] can be written in a slope-intercept form as [tex]y = 4-\dfrac{x}{2}[/tex].

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The first three values in a selected series are 3, 6, and 9. what are the next three va;ies that will be inserted by autofill?

Answers

the answer is 12, 15. and 18

Answer:

the answer is 12, 15. and 18.

Step-by-step explanation:

You purchase three packs of batteries at $11.99/pack. You have a coupon for 10% off and the sale tax is 7% how much is the total cost of your purchase?

Answers

The answer is 34.64. C.

The entire purchase of the three items cost about $34.

How to determine the total cost?

The given parameters are:

The cost of one packs of batteries is $11.99/pack.

Coupon = 10% off

The total cost is then calculated using:

Total = (three packs of batteries) * (1 - Coupon)

So, we have:

Total = (11.99 + 11.99 + 11.99 ) * (1 - 10%)

Evaluate,

Total = 34.47

Approximate

Total = 34

Hence, the entire purchase of the three items cost about $34.

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a square has the side measurment of s = 2 3 over 2 using the formula 1= s^2 find the area of the square

Answers

check the picture below.

[tex]\bf A=s^2\qquad s=2\frac{3}{2}\qquad A=\left( 2\frac{3}{2} \right)^2\implies A=\left( \frac{2\cdot 2+3}{2} \right)^2 \\\\\\ A=\left( \frac{7}{2} \right)^2\implies A=\cfrac{7^2}{2^2}\implies A=\cfrac{49}{4}\implies A=12\frac{1}{4}[/tex]

X/2-18=(-28)
What is x?
X=?

Answers

X = -20

Step 1: Write the division as a fraction
x ÷ 2 - 18 --> x/2 - 18

Step 2: Multiply both sides of the equation by 2
x - 36 = -56

Step 3: Move contact to the right side and change its sign
x = -56 +36

Step 4: Solve
x = -20

X would be -20 in this problem.

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