Given the curve [tex]\displaystyle \large{y=e^{-x}\sin x \ \ (x\geq 0)[/tex] let the area enclosed by the curve and the x-axis (above the x-axis) be S0, S1, S2, ... , Sn, ... in order from y-axis. Find [tex]\displaystyle \large{ \lim_{n \to \infty} \sum_{k=0}^{n} S_k}[/tex]

Answers

Answer 1

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]


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Answers

Answer:

37, 79, 7

Step-by-step explanation:

f(6) = 6² -6 +7

36-6+7 is 37

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Answers

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

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Answers

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

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Given y = mx + b

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last option in the picture

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Answer:

-c-4≥8

Step-by-step explanation:

Just ask!

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Answer:

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Step-by-step ex planation:

The rate of change is just a fancy word for slope!

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Step-by-step explanation:

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Answers

Answer:

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Step-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

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Brendan says that the negative of - 1/3
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​Is Brendan correct? Explain your answer.

Answers

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.

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Negative "of" a number is the same number with opposite sign.

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-1/3 and 3 do not lie at the same distance from zero.

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Therefore, Brendan is wrong as the negative of a number has opposite sign but same "magnitude".

Solve.

x2=−0.73

What is the solution to the equation?

Enter your answer as a decimal in the box.

x =

Answers

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

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Answer: Did you copy and paste this because this just hits different. message me if i misunderstood this.

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?

Answers

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Step-by-step explanation:

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Answers

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Answers

The answer is the following

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Problem #3
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Answers

Answer:

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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

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OA) $316.47 and $372.64

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Step-by-step explanation:

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)}

Answers

Answer:

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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:

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Answers

Answer: y = -x - 3

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Hi there! :)

5-2b when b=-1; 3.

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~Answer given by

An Unknown Dreamer

Good luck.

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