16<br> If t> 0 and +2-4 = 0, what is the value of ' ?

Consider the following line integral. C xy dx + x2y3 dy, c is counterclockwise around the triangle with vertices (0, 0), (1, 0),

Sedaia [141]

It looks like the integral is

$\displaystyle \int_C xy \, dx + x^2y^3 \, dy$

Let's close the loop by adding a line integral over the line segment joining (1, 4) to (0, 0). Then the closed loop is the triangular region

T = {(x, y) : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 4x}

Since the integrand has no singularities over or on the boundary of T, we have by Green's theorem

$\displaystyle \int_C xy \, dx + x^2y^3 \, dy = \iint_T \left(\frac{\partial(x^2y^3)}{\partial x} - \frac{\partial(xy)}{\partial y}\right) \, dx \, dy = \int_0^1 \int_0^{4x} (2xy^3 - x) \, dy \, dx$

Compute the double integral:

$\displaystyle \int_0^1 \int_0^{4x} (2xy^3 - x) \, dy \, dx = \int_0^1 \int_0^{4x} \left(128x^5 - 4x^2\right) \, dx = \frac{60}3$

From this result, we subtract the line integral over the extra line segment we added. Parameterize this path by

C' : {(1 - t, 4 - 4t) : 0 ≤ t ≤ 1}

The line integral over C' is

$\displaystyle \int_{C'} xy \, dx + x^2y^3 \, dy = \int_0^1 (1-t)(4-4t)(-dt) + (1-t)^2(4-4t)^3(-4\,dt)$

$\displaystyle \int_{C'} xy \, dx + x^2y^3 \, dy = \int_0^1 \left(256 t^5-1280 t^4+2560 t^3-2564 t^2+1288 t-260 \right) \, dt = -44$

so that the line integral over C alone is 60/3 - (-44) = 64.

3 0

3 years ago

Speeding ticket Littering fine Library fine Detention All of these are examples of

wlad13 [49]

Broken laws
because each one of them you're doing some thing wrong

3 0

3 years ago

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After Li Shimin ascended the throne, he actively listened to the opinions of his ministers, ruled the world with civility, and e

Monica [59]

Answer:

Tang Taizong Li Shimin (January 28, 598 - July 10, 649) was the second emperor of the Tang Dynasty. he was honored as the "Khan of Heaven" by the nine surnames Tiele, the king of the Western Regions, and the guardian of Tochara.

Explanation:

the king of the Western Regions, and the guardian of Tochara.

3 0

2 years ago

Kysssssssssssssssssssssssssssssssssssssssssssssssssss

Firlakuza [10]

<h3><u>PLEASE</u><u> MARK</u><u> ME</u><u> BRAINLIEST</u><u> </u>❤️✌</h3>

7 0

3 years ago

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True or false: Gender-typing is fueled by gender schemas. True false question.

LekaFEV [45]

It is True that Gender-typing is fueled by gender schemas.

<h3>What is Gender schema?</h3>

Gender schema theory serves as cognitive theory of gender development that stress that any gender serves as output of a particular culture.

Gender schema theory can be attributed to

psychologist Sandra Bem who gave an assertion about how children is able to continuously learn about male as well as female roles from the culture that they grow up with.

This theory stressed that children can easily adjust their behavior so as to be able to fit in with the gender norms as regards their culture even from lowest stages of social development.

The theory was put up by psychologist Sandra and he explains that people process information, in part.

Learn more about gender schemas at

brainly.com/question/7277651

#SPJ1

5 0

2 years ago

16 If t> 0 and +2-4 = 0, what is the value of ' ?