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]
Find the product z1z2 in standard form. (Simplify your answer completely.)
z1 = 9 + 3i, z2 = −9 + 3i
Answer:
See image
Step-by-step explanation:
These are "complex numbers" because they have the imaginary number, i, in the expression.
The 9 (and the -9) is the real part and the 3i is the imaginary part. But when you add or subtract, multiply or divide them you can manipulate them just as if the i was a variable (like x or n or y) If you learned to multiply
(x+2)(x+5)
= x^2 + 7x + 10
This problem will work the same way. You may have learned "FOIL" or using a box to multiply two binomials (two-piece things like x+2)
**SUPER IMPORTANT NOTE: i^2 is equal to -1
These particular complex numbers are called conjugates which means they have the same 'pieces' the 9 and the 3i, with only one sign different. So there is a short cut for this problem and the imaginary number falls out and we get a real number answer. This doesn't happen for every question, only when you multiply conjugates, but the method shown here works for multiplying complex numbers all the time.
What is the value of the expression Negative 225 divided b (negative 15)?
What is the value of the expression 30 divided by (negative 6)?
Answer:
-15-5Step-by-step explanation:
What is the value of the expression Negative 225 divided b (negative 15)?
What is the value of the expression 30 divided by (negative 6)?
I answer for what I understand
-225 : 15 = - 15
30 : (- 6 ) = -5
the length of 2x metres and it's width is 2 metres shorter than its length. what is the area of the rectangle
Answer:
4x^2-4x
Step-by-step explanation:
2x*(2x-2)
(2x)(2x)+(2x)(−2)
[tex]4x^2-4x[/tex]
Hope this helped! :)
please help me with math
Hi there! :)
5-2b when b=-1; 3.
5-2(-1)
5+2
7
5-2(3)
5-6
-1
3(x+8)
3x+24
Hope it helps!
~Answer given by
An Unknown Dreamer
Good luck.
my Christmas present to you
Answer:
super man
Step-by-step explanation:
if ur good at math, pls help. will get brainliest
Answer: y = -x - 3
Using the parent function/formula y = mx +b, you can use the slope formula [tex]\frac{x_{2}-x_1 }{y_2-y_1}[/tex] to determine that the slope is negative. Then, using the answer to the slope equation (m), you can use 2 points from the table to solve the equation for b. Then just write the equation using y, x, value of m, and value of b.
Simply using quadratic formula! Brainliest will be awarded if correct
Answer:
[tex]\left \{ {{y1=-1.2+0.4i} \atop {y2=-1.2-0.4i}} \right.[/tex]
Step-by-step explanation:
5y + 12 = [tex]\frac{-8}{y}[/tex]
y*(5y+12)=-8
5[tex]y^{2}[/tex] + 12y = -8
5[tex]y^{2}[/tex] + 12y + 8 = 0
Δ(Delta) = [tex]b^{2}[/tex] - 4ac
= [tex]12^{2}[/tex] - 4*5*8
= -16
Δ<0
therefore we have complex number
y = [tex]\frac{-b}{2a}[/tex] ± [tex]\frac{\sqrt{|Delta|} }{2a}[/tex]
[tex]\left \{ {{y1=-1.2+0.4i} \atop {y2=-1.2-0.4i}} \right.[/tex]
Brendan says that the negative of - 1/3
is 3 because they are on opposite sides of zero when they are plotted on the number line.
Is Brendan correct? Explain your answer.
Answer:
No! Brendan has totally got the wrong idea.
Step-by-step explanation:
Numbers on the right hand side of zero are said to be positive, while, the numbers on the opposite side are said to be negative.
That means Brendan's correct in saying that -1/3 lies on the opposite side of 3, since negative numbers lie on the opposite side of the positives.
But,
Negative "of" a number is the same number with opposite sign.
That is, if a number is 3 units away from the zero, it's negative will also be 3 units away from zero, but on the opposite side.
-1/3 and 3 do not lie at the same distance from zero.
-1/3 would be approached before -1 by walking in the negative direction whereas 3 would be approached after 1.
Therefore, Brendan is wrong as the negative of a number has opposite sign but same "magnitude".
Can someone plss help me with this!!!
Answer:
10/1 or 10
Step-by-step ex planation:
The rate of change is just a fancy word for slope!
To find the slope of this line I found two points of the graph that were easy to determine. Slope = rise/run
which is this case was rise 10 because the point went from 20 to 30 and the run being 1 because it was going from 1 to 2
Hope this helps!!
Please tell me if I have messed up!!
I enjoy learning from my mistakes:)
Plot /\ABC: B(-1,-1), and C(4,-1)
——JUST LOOK AT PHOTO AND PLEASE HELP
Answer:
i dont know sry
Step-by-step explanation:
:)
what is 9 less than 3 times f
Use long division to find the quotient below. (8x^3-10x^2-19x-15)/(2x-5)
Answer:
4x2 + 5x + 3
Step-by-step explanation:
What will be the result when each term sin2theta+cos2theta =1 is divided by sin2theta
Answer: Did you copy and paste this because this just hits different. message me if i misunderstood this.
Step-by-step explanation:
Hey y’all,please help me :)
Answer:
-c-4≥8
Step-by-step explanation:
Just ask!
Hope this helps. :)
find the equation of the line with slope of 2 passing through (3,1)
Answer:
y = 2x - 5
Step-by-step explanation:
Problem #3
Imagine you are building a triangular garden. Two sides of the garden measure 230 ft and
360 ft and the included angle between these two sides of the garden measures 38°. What is
the measure of the third side?
С
230ft
38
360ft
Answer:
b = 115.4 feet
Step-by-step explanation:
b^2 = a^2 + c^2 - 2 (a)(b) x cos B
b^2 = 230^2 + 360^2 - 2(230)(360) x cos(38 degrees)
b^2 = 52,900 + 129,600 - 2(230)(360) x cos(38 degrees)
b^2 = 182,500 - 165,600 x cos(38 degrees)
b = [tex]\sqrt{(16,900)(cos(38)) }[/tex]
b = 115.4 feet
i hope this helps!! :)
Answer:
115.4 ft
Step-by-step explanation:
find the side of square whose perimeter. is 24.8cm.
Answer:
6.2
Step-by-step explanation:
What function equation is represented by the graph?
Answer:
f(x) = (2/3)x-2
Step-by-step explanation:
Pick any two points and calculate the slope. I chose: (-3,-4) and (3,0). The Rise/Run, or slope, is (0-(-4))/(3-(-3) or 4/6: 2/3
The y-intercept of the graph is -2 ( the value of y when x=0).
Given y = mx + b
we have
y = (2/3)x - 2
Option 3
aaaaaaaaaaaaaaaaaaaa
Answer:
what grade are you??..........
please help asap!!!!
Answer:
justin biberis black
Step-by-step explanation:
Solve the following system of equations.
2x + y=3
x=2y-1
Make sure there are NO SPACES in your answer. Include a comma in your answer.
ANSWER: {(
)}
Answer:
x=1,y=1
Step-by-step explanation:
For this problem, we can use the substitution method. Since we know that x is equal to 2y-1, we can rewrite the first equation as 2(2y-1)+y=3. Now solving we get:
4y-2+y=3
5y-2=3
5y=5
y=1
Now we can plug in 1 for y in the second equation and get:
x=2(1)-1
x=1
This means that x=1, and y=1
Help me pls imma fail
Answer:
-
Step-by-step explanation:
Calculate the frequency of a wave with a speed of 130 m/s and a wavelength of 2.5
Answer:
52 Hz
Step-by-step explanation:
We can divide the speed by the wavelength to find the number of waves propagated in one second:
(130 m/s)/(2.5 m/wave) = 52 waves/s
The SI unit of frequency is the Hertz, equal to one cycle per second.
The frequency of this wave is 52 Hz.
_____
Additional comment
The use of the unit Hz to signify frequency was adopted in 1933, but did not replace the unit "cycles per second" (CPS) until a number of years later.
FLOORING Miguel is putting hardwood flooring in a hallway that measures 4 feet by 17 feet. The wood he has chosen is $5.48 per square foot
What is the range of the actual cost of putting the flooring in the hallway?
OA) $316.47 and $372.64
OB) $372.64 and $431.55
O C) $362.64 and $382.64
OD) $316.47 and $431.55
A I think yup it's A good luck
Step-by-step explanation:
Miguel is putting hardwood flooring in a hallway that measures 4 feet by 17 feet. The wood he has chosen is $5.48 per square foot
What is the range of the actual cost of putting the flooring in the hallway?
OA) $316.47 and $372.64
OB) $372.64 and $431.55
O C) $362.64 and $382.64
OD) $316.47 and $431.55
Write the following equation in standard form:`X^{2}+6X=-2`
Answer:
[tex]x^2 +6x +2=0[/tex]
Step-by-step explanation:
Tommy bought one dozen donuts. He gave one-sixth of the donuts to his wife. He then gave one-fifth of the remaining donuts to his son. He took the rest to work. How many donuts did Tommy take to work?
Answer:
8
Step-by-step explanation:
1/6 of 12 is 2
12 - 2 = 10
1/5 of 10 is also 2
10 - 2 = 8
Tommy took 8 donuts to work.
Given the function f(x) = x2 - x + 7, find each of the following.
f(6)=
f(- 9)=
f(0)
Answer:
37, 79, 7
Step-by-step explanation:
f(6) = 6² -6 +7
36-6+7 is 37
f(-9) = -9² -9+7
81-9+7 is 79
f(0) = 0² -0+7
0-0+7 is 7
What inequality is represented by this graph?
O X > 6 O X _< 6 O X<6 OX_>6
last option in the picture
Find the area of a trapezoid with height h and bases b1 and b2.
h=6 in.
b1=9 in.
b2=11 in.
Solve.
x2=−0.73
What is the solution to the equation?
Enter your answer as a decimal in the box.
x =
x = -0.365 (If its x(2)=-0.73)
How I solved it:
x^2=-0.73
-0.73÷2=-0.365
//
-0.365×2=-0.73
Therefore, the answer is -0.365.
Answer:
x = -1.46 (if its x/2=-0.73)
x/2=-0.73
-0.73x2=-1.46
therefore,
-1.46/2=-0.73
///
making the answer
-1.46