Evaluate ∫C ysin(z)ds, where C is the circular helix given by the equations x = cos(t), y = sin(t), z = t, 0 ≤ t ≤ 2π. SOLUTION The formula for a line integral in space gives the following. ∫y sin(z)ds = sin2(t) dt = (sin(t))2√ (cos(t))2 + (sin(t))2 + 1dt = 1 2 (1 - cos(2t))dt = √2 2 =

Answers

Answer 1

Answer:

The line integral is

$\displaystyle\int_Cy\sin z\,\mathrm ds=\int_0^(2\pi)y(t)\sin z(t)\,\sqrt{\left((\mathrm dx)/(\mathrm dt)\right)^2+\left((\mathrm dy)/(\mathrm dt)\right)^2+\left((\mathrm dz)/(\mathrm dt)\right)^2}\,\mathrm dt$

We have

$x=\cos t\implies(\mathrm dx)/(\mathrm dt)=-\sin t$

$y=\sin t\implies(\mathrm dy)/(\mathrm dt)=\cos t$

$z=t\implies(\mathrm dz)/(\mathrm dt)=1$

so the integral reduces to

$\displaystyle\int_0^(2\pi)\sin^2t√((-\sin t)^2+\cos^2t+1^2)\,\mathrm dt=\frac{\sqrt2}2\int_0^(2\pi)(1-\cos2t)\,\mathrm dt=\boxed{\frac\pi{\sqrt2}}$

Answer 2

Answer:

The line integral ∫C ysin(z) ds over the circular helix C, parametrized by x = cos(t), y = sin(t), z = t for 0 ≤ t ≤ 2π, evaluates to π√2.

To evaluate the line integral ∫C ysin(z) ds over the circular helix C given by x = cos(t), y = sin(t), z = t for 0 ≤ t ≤ 2π, we follow these steps:

1. Parameterize the curve: C is already parameterized as x = cos(t), y = sin(t), z = t.

2. Find the differential ds: ds = √(dx² + dy² + dz²) = √(sin²(t) + cos²(t) + 1)dt = √(1 + 1)dt = √2 dt.

3. Evaluate the integral: ∫C ysin(z) ds = ∫[0, 2π] sin(t) * sin(t) * √2 dt = ∫[0, 2π] sin²(t) * √2 dt.

Now, we'll integrate sin²(t) * √2 with respect to t:

∫ sin²(t) * √2 dt = (1/2) * ∫ (1 - cos(2t)) * √2 dt.

Using the power rule for integration, we get:

(1/2) * [(t - (1/2) * sin(2t)) * √2] | [0, 2π].

Plugging in the limits:

(1/2) * [(2π - (1/2) * sin(4π) - (0 - (1/2) * sin(0))) * √2].

Since sin(4π) = sin(0) = 0:

(1/2) * [(2π - 0 - 0) * √2] = π√2.

So, ∫C ysin(z) ds = π√2.

For more such questions on Line Integral :

brainly.com/question/28525062

#SPJ6

Related Questions

-2√7 rational or irrational?
C^2 + 5c =3 Tell if this factors or not? And show how it factors or does not factor? And if it factors what are the roots?
If there are 100 cm in1 meter, how manycentimeters are in 6meters?
Find the area of this circle. Use 3 for pi
Which procedure would you use to convert a fraction to a decimal?1Convert to a decimal.5to a decimal?

Write this fraction in the simplest form 15/25

Answers

15/25 Divididing numerator and denominator by 5 we get:

3 / 5

If 15/25 is 0.6 you can then convert it to a 3/5

1011
For questions 6-8, determine the number of solutions.
Type your answers as one, none, or many (no capital letters)
6
4(x + 5) - 2 = 18 + 4x

Answers

the answer to that would be many

Please let me know. This is 2nd grade math.'how is exercise 5 different from exercise 6 explain.'

Answers

Answer:

Show a photo

Step-by-step explanation:

we can't see what you're talking about

Can you show a photo of your question? Because I don't know how to answer that.

16 1/9+ -4 2/3 help please??

Answers

Answer:

11 4/9

Step-by-step explanation:

ANDDD the answer is

A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationshipbetween the given line segment and its image is true?
A The line segments are parallel, and the image is twice the length of the given line segment.
B. The line segments are parallel, and the image is one-half of the length of the given line segment.
C. The line segments are perpendicular, and the image is twice the length of the given line segment.
DD The line segments are perpendicular, and the image is one-half of the length of the given line segment.

Answers

9514 1404 393

Answer:

A The line segments are parallel, and the image is twice the length of the given line segment.

Step-by-step explanation:

Dilation by a factor of 2 means any measure of the image is 2 times the corresponding measure of the original.

Dilation does not change any orientations, so the image will have the same orientation with respect to the origin, axes, or any other line segments. That means the dilated segment is parallel to the original. (If the center of dilation is on the original line segment, the dilated segment will overlay the original segment. That is specifically not the case here.)

Which expression means the same as subtract 3 from 7 and then multiply by 4

Answers

7-3x4 subtract 3 from 7 means you need to take 3 out of 7 (7-3) then you need to multiply that by 4!

Hope this helped!

Other Questions

Which is the solution to the system of equations?y = 1/8x - 1-5x + 4y = -13A. (0, -1)B. (8,0)C. (1, -7/8)D. (2, -3/4)

GUYS HELP ME PLS. Tina is standing at the bottom of a hill. Matt is standing on the hill so that when Tina's line of sight isperpendicular to her body, she is looking at Matt's shoes.a. If Tina's eyes are 5 feet from the ground and 14.5 feet from Matt's shoes, what is the angle of elevation ofthe hill to the nearest degree?

Please help me understand I am confused

From data gathered in the period 2008−2012, the yearly value of U.S. exports can be modeled by the function E(x) = −228x3 + 2,252.8x2 − 6,098.5x + 11,425.8 where x is the number of years after 2008 and E(x) is the value of exports in billions of dollars. The yearly value of U.S. imports can be modeled by the function I(x) = −400.4x3 + 3,954.4x2 − 11,128.8x + 17,749.6 where x is the number of years after 2008 and I(x) is the value of imports in billions of dollars. Estimate the total value the U.S. imported and exported in 2011. Round to the nearest whole number. About billion dollars were imported and exported by the U.S. in 2011.