The given curve crosses the x-axis whenever x is a multiple of π, and it lies above the x-axis between consecutive even and odd multiples of π. So the regions with area S₀, S₁, S₂, ... are the sets
[tex]R_0 = \left\{(x, y) : 0 \le x \le \pi \text{ and } 0 \le y \le e^{-x}\sin(x)\right\}[/tex]
[tex]R_1 = \left\{(x, y) : 2\pi \le x \le 3\pi \text{ and } 0 \le y \le e^{-x}\sin(x)\right\}[/tex]
[tex]R_2 = \left\{(x, y) : 4\pi \le x \le 5\pi \text{ and } 0 \le y \le e^{-x}\sin(x)\right\}[/tex]
and so on, with
[tex]R_k = \left\{(x, y) : 2k\pi \le x \le (2k+1)\pi \text{ and } 0 \le y \le e^{-x}\sin(x)\right\}[/tex]
for natural number k.
The areas themselves are then given by the integral
[tex]S_k = \displaystyle \int_{2k\pi}^{(2k+1)\pi} \int_0^{e^{-x}\sin(x)} dy \, dx = \int_{2k\pi}^{(2k+1)\pi} e^{-x}\sin(x) \, dx[/tex]
Integrate by parts twice. Take
[tex]u = e^{-x} \implies du = -e^{-x} \, dx[/tex]
[tex]dv = \sin(x) \, dx \implies v = -\cos(x)[/tex]
so that
[tex]\displaystyle \int e^{-x}\sin(x) \, dx = -e^{-x}\cos(x) - \int e^{-x}\cos(x) \, dx[/tex]
then
[tex]u = e^{-x} \implies du = -e^{-x} \, dx[/tex]
[tex]dv = \cos(x) \, dx \implies v = \sin(x)[/tex]
so that
[tex]\displaystyle \int e^{-x}\cos(x) \, dx = e^{-x}\sin(x) + \int e^{-x}\sin(x) \, dx[/tex]
Overall, we find
[tex]\displaystyle \int e^{-x}\sin(x) \, dx = -e^{-x}\cos(x) - e^{-x}\sin(x) - \int e^{-x}\sin(x) \, dx[/tex]
or
[tex]\displaystyle \int e^{-x}\sin(x) \, dx = -\frac12 e^{-x} (\cos(x)+\sin(x)) + C[/tex]
Using the antiderivative and the fundamental theorem of calculus, we compute the k-th area to be
[tex]\displaystyle S_k = -\frac12 e^{-(2k+1)\pi} (\cos((2k+1)\pi)+\sin((2k+1)\pi)) + \frac12 e^{-2k\pi} (\cos(2k\pi)+\sin(2k\pi))[/tex]
[tex]\displaystyle S_k = \frac12 e^{-2k\pi} \cos(2k\pi) \left(e^{-\pi} + 1\right)[/tex]
[tex]\displaystyle S_k = \frac{e^{-\pi}+1}2 e^{-2k\pi}[/tex]
Since [tex]\left|e^{-2\pi}\right|<1[/tex], the sum we want is a convergent geometric sum. As n goes to ∞, we have
[tex]\displaystyle \lim_{n\to\infty} \sum_{k=0}^n S_k = \frac{e^{-\pi}+1}2 \cdot \frac1{1 - e^{-2\pi}} = \boxed{\frac{e^\pi+1}{4\sinh(\pi)}}[/tex]
Which of the following can be a common denominator for the fractions 6/7 and3/4 ? 14, 4, 7, 28
Show two ways three people could share a 5 segment chewy worm
Answer:
Each person would get 1/5 of 3 segments.
Each person would get 3/5 of a segment.
Step-by-step explanation:
(3) (1/5) = 3/5 segments
A math student goes to an office supply store and purchases a total of 16 spirals and pencils. Each spiral costs $3. 50, and each pencil costs $1. 25. If the student spent a total of $29, how many pencils did the student purchase?.
The student purchase 12 pencils
Let the total number of pencils purchased be xLet the total number of spirals purchased be yIf a math student goes to an office supply store and purchases a total of 16 spirals and pencils, then;
x + y = 16 .................. 1
If each spiral costs $3. 50, and each pencil costs $1. 25 and the student spent a total of $29, hence;
1.25x + 3.5y = 29 ................. 2
Solve both equations simultaneously
x + y = 16 .................. 1 * 1.25
1.25x + 3.5y = 29 ................. 2 * 1
__________________________________
1.25x + 1.25y = 20.
1.25x + 3.5y = 29
Subtract both equations
1.25y - 3.5y = 20 - 29
-2.25y = -9
y = 9/2.25
y = 4
Substitute y = 4 into equation 1:
x + y = 16
x = 16 - 4
x = 12
This means that the student purchase 12 pencils
Learn more on simultaneous equations here: https://brainly.com/question/15165519
Help me please just help I need help rn like rn
Answer:
i.900
j.6
k.8
l.0
Step-by-step explanation:
hope this helps you ;)
can you please answer this question asap
x-1/3-x+3/4=-1
The value of x is [tex]\mathbf{8\dfrac{1}{3}}[/tex]
The objective of this question is to evaluate and determine the value of x. The fraction given in the question consists of algebraic expressions and, we need to solve it through following process;
∴
[tex]\mathbf{\dfrac{x-1}{3-x} + \dfrac{3}{4}=-1 }[/tex]
Collect like terms
[tex]\mathbf{(\dfrac{x-1}{3-x})=-1 -\dfrac{3}{4}}}[/tex]
[tex]\mathbf{(\dfrac{x-1}{3-x})=-(\dfrac{7}{4}})}[/tex]
-4(x - 1) = -7(3 -x)
-4x + 4 = -21 + 7x
4x -7x = -21 - 4
-3x = - 25
[tex]\mathbf{x= \dfrac{25}{3}}[/tex]
[tex]\mathbf{x= 8\dfrac{1}{3}}[/tex]
Learn more about evaluating fractions here:
https://brainly.com/question/12023322
NEED HELP ASAP WILL MARK BRAIN <3
A cash card has a starting value of $25. If the card is not used within the first year of its purchase, the value on the card begins to decrease by $2.50 per month.
The relationship between the number of months after the first year and the amount remaining on the card is .
The independent variable is the _____________?
The dependent variable is the ______________?
Solve each inequality
2x + 2 - 6x > -4 - 7x
1 + 7n ≤ 15 or -2n - 2 ≤ - 18
Please show all work :D
Answer:
x > -2n ≤ -2 or n ≥ 8Step-by-step explanation:
1.Add 7x-2 to both sides and collect terms.
2x +2 -6x +7x -2 > -4 -7x +7x -2
3x > -6
x > -2 . . . . . . . divide by 3
__
2.Solve these one at a time, and form the union of the answers.
1 +7n ≤ 15 . . . .given
7n ≤ 14 . . . . . . subtract 1
n ≤ 2 . . . . . . . divide by 7
__
-2n -2 ≤ -18 . . . given
-2n ≤ -16 . . . . . add 2
n ≥ 8 . . . . . . . . divide by -2
The solution is n ≤ -2 or n ≥ 8.
Which of the expressions below has a product that is greater
than 140? Select all that apply.
A) 11.5 x 12
B) 11.85 x 12
C) 15.2 x 9
D) 23.4 x 6
You can earn 5 coins
Find the slope of the following graph and write ur results in the empty box.
Marta bought c CDs for $ 12 each, b books for $ 7 each, and a bag for $ 24 to.
Write an expression to show the total money Marta spent.
If Marta bought 4 CDs and 3 books, how much did she spend?
Answer:
A, 12c + 7b + 24
B, $69
Step-by-step explanation:
Please, read my picture Thanks! Also tell me if its right in the comment
4 * 12 + 3 * 7 + 24 = m
m represents how much money Marta spent
48 + 21 + 24 = m
69 + 24 = m
$93 = m
Bro this is so hard
Answer:
c = 50 degrees and d = 30 degrees
Answer:
C = 80 degrees
D = 30 degrees
Step-by-step explanation:
To find the angle of C you have to add the other angles of that triangle. The total angle of a triangle is 180 degrees.
50 + 30 = 80
Subtracted 80 from 180 to get the other missing angle next to angle c. That is 100. A straight line is 180 degrees so you get 80 degrees for angle C.
To find the angle of D you add angle C, which is 80, to the 100 degree angle and get 180. The total degrees for a four-sided shape is 360 degrees. Then you try to figure out the other missing angle next to the 30-degree angle. A straight line is 180 degrees so subtract 30 from 180 to get the value of the missing angle.
180 - 30 = 150
Add the 180 degrees from earlier to 150 and you get 330. Subtract 330 from 360 and you get 30 degrees. Therefore the angle for d is 30 degrees.
factor the expression 9x^2+4 explain your steps if not possible explain why not
A basketball player only scores two-point and three point goals during a game. The basketball player score a total of 25 points with a total of 11 goals during the game
Answer:
3 three pointers and 8 two pointers
Step-by-step explanation:
hope this helps
Solve using the Quadratic Formula:
a) -7x^2 + 7x + 1 = -8
b) -6x - 1 + 5x^2 = 8x^2
Show your work.
A. - 7x² + 7x + 1 + 8 = 0
- 7x² + 7x + 9 = 0
x = - b ± √b² - 4ac / 2a
= - 7 ± √7² - 4 (-7) (9) / 2 (-7)
= - 7 ± √49 + 252 / - 14
= - 7 ± √ 301 / - 14
x = - 7 + 17.349 / - 14 x = - 7 - 17.349 / - 14
= 10.349 / - 14 = - 24.349 / - 14
= - 0.739 = - 1.739
B. 8x² - 5x² + 6x + 1 = 0
3x² + 6x + 1 = 0
x = - b ± √b² - 4ac / 2a
= - (-5) ± √-5² - 4 (3) (1) / 2 (3)
= 5 ± √25 - 12 / 6
= 5 ± √ 13 / 6
x = 5 + 3.606 /6 x = - 5 - 3.606 / 6
= 8.606/6 = - 8.606/6
= 1.434. = - 1.434
please re-check the answers hope this helps
write the equation of the line in fully simplified slope-intercept form
y= -4x-5 you can find this by seeing how far it goes up/down and where it starts:)
You might need: 3 Calculator
2
Find the area of the shape.
Either enter an exact answer in terms of t or use 3.14 for it and enter your answer as a decimal.
units?
Answer:
3.14 or pi
Step-by-step explanation:
the area of a circle = [tex]pi * r^{2}[/tex] where r is the radius of the circle
this is 1/4 of the circle, so it would be 1/4 (pi * r²)
it shows that the radius is 2, so you have
1/4 ( pi * 2²)
= 1/4 ( pi * 4 )
= 1/4 (3.14 * 4)
= 3.14 units²
Please help meeee! Need the answer for the first box, no phony answers please! Thanks for your help, and happy holidays!
Answer:
y=2(x+5)^2-4.
Step-by-step explanation:
By solving a little bit, we get the vertex to be (-5, -4). Plugging the info given and the vertex into the vertex form, we get y=2(x+5)^2-4. You have a happy holiday season, too!
What is the Distance formula of (-6, -12) and (4,8)
Distance formula is in picture
Answer:
The distance formula is 22.360679775
if you round it to the nearest tenth, then the answer is 22.4
Step-by-step explanation:
(-6, -12) and (4,8)
(x2 - x1)² + (y2 - y1)²
Place the x and y values
(4 - - 6)² + (8 - - 12)²
= 10² + 20²
= 100 + 400 = 500
√500 = 22.360679775
Round it to the nearest tenth and you will get 22.4
Hope this helps!
please mark me as the brainliest
The formula in finding the distance between two points is given below:
[tex] \sf{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}[/tex]where:
[tex] \sf{(x_1, y_1) = (-6, -12)}[/tex][tex] \sf{(x_2, y_2) = (4, 8)}[/tex]Substitute the given values into the distance formula and solve for d:
[tex] \sf{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}[/tex][tex] \sf{d = \sqrt{[4 - ( - 6)]^2 + [8 - ( - 12)]^2}}[/tex][tex] \sf{d = \sqrt{(10)^2 + (20)^2}}[/tex][tex] \sf{d = \sqrt{100 + 400}}[/tex][tex] \sf{d = \sqrt{500}}[/tex][tex] \bold{d = 22.4 \: units}[/tex]Therefore, the distance between the two points is 22.4 units (nearest tenth).
Cheers!
How many lines of symmetry does the figure have?
0
6
4
5
please help! the ppl who are good atttt mathhh please!
Answer:
10) SAS
12) SAS
13) SSS
11) ASA
14) ASA
Answer:
10. none
11. AAS
12. SAS
13. SSS
14. SAS
15. ASA
Step-by-step explanation:
I'm not going into detail of each question... I have a 101 in geometry right now though and I've already done triangles, so just trust me on this one.
This graph shows the amount of (s) money in Jack's account after (w) weeks.
Which equation represents the amount of (s) money in Jack's account after (w) weeks?
Answer:
s=6w
Step-by-step explanation:
the graph goes up by 6 on the y axis for everyone one that the x moves
Answer:
d
Step-by-step explanation:
HELP ME PLEASEEE !!
DON'T GIVE ME IRRELEVANT ANSWERS PLEASE
Find the product
(-2) (7)
Answer:
-14
Step-by-step explanation:
Remember that a negative times a positive is negative:
(-2)(7) = -(2)(7) = -14
i will mark brainlist !!!!!!!!
Answer:
f(n)=4*2^(n-1)
f(n)=a*b^n -> 2 is the multiplier for the outputs, so b=2. 4 is the output when n=1 (all answers are to the power of n-1), so a=4
Plz answer rn nmmmm ASAP!!
Answer:
C
Step-by-step explanation:
a
Fitness Club A has a membership fee of $35 plus $3 each time you swim. Fitness Club B has a membership fee of
$27.50 plus $4.50 each time you swim. After how many swims will the cost be the same?
A
4
B) 7
6
D
8
E 5
Write an equation for each one set the equal to each other and solve.
A: 35 + 3x
B: 27.50 + 4.50x
35 + 3x = 27.50 + 4.50x
Subtract 27.50 from both sides:
7.50 + 3x = 4.50x
Subtract 3x from both sides
7.50 = 1.50x
Divide both sides by 1.50
X = 5
Answer: 5
A. Express the following in scientific notation 1.0.948901 2.11,000,000 3.0.08907
Answer:
answer should be c good luck
Xavier's pet turtle laid eggs! So far, only 2 eggs have hatched for every 5 that are unhatched. If Xavier's turtle laid 35 eggs in all, how many eggs have hatched?
Answer: 21
Step-by-step explanation:
plass help brainlist if possible compare 9/10 and 7/5
Answer:
9/10 < 7/5
Step-by-step explanation:
7/5 = 14/10
9/10 < 14/10
Answer:
7/5 is greater than 9/10
Step-by-step explanation:
7/5 = 14/10
Calculator
What is the value of x
Answer:
71.8
Step-by-step explanation: