Sabrina has designed a rectangular painting that measures 60 feet in length and 40 feet in

use green's theorem to evaluate the line integral along the given positively oriented curve. c 9y3 dx − 9x3 dy, c is the circle

Rina8888 [55]

The line integral along the given positively oriented curve is -216π. Using green's theorem, the required value is calculated.

<h3>What is green's theorem?</h3>

The theorem states that,

$\int_CPdx+Qdy = \int\int_D(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})dx dy$

Where C is the curve.

<h3>Calculation:</h3>

The given line integral is

$\int_C9y^3dx-9x^3dy$

Where curve C is a circle x² + y² = 4;

Applying green's theorem,

P = 9y³; Q = -9x³

Then,

$\frac{\partial P}{\partial y} = \frac{\partial 9y^3}{\partial y} = 27y^2$

$\frac{\partial Q}{\partial x} = \frac{\partial -9x^3}{\partial x} = 27x^2$

$\int_C9y^3dx-9x^3dy = \int\int_D(-27x^2 - 27y^2)dx dy$

⇒ $-27\int\int_D(x^2 + y^2)dx dy$

Since it is given that the curve is a circle i.e., x² + y² = 2², then changing the limits as

0 ≤ r ≤ 2; and 0 ≤ θ ≤ 2π

Then the integral becomes

$-27\int\limits^{2\pi}_0\int\limits^2_0r^2. r dr d\theta$

⇒ $-27\int\limits^{2\pi}_0\int\limits^2_0 r^3dr d\theta$

⇒ $-27\int\limits^{2\pi}_0 (r^4/4)|_0^2 d\theta$

⇒ $-27\int\limits^{2\pi}_0 (16/4) d\theta$

⇒ $-108\int\limits^{2\pi}_0 d\theta$

⇒ $-108[2\pi - 0]$

⇒ -216π

Therefore, the required value is -216π.

Learn more about green's theorem here:

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3 0

2 years ago

Help me I need help please anyone

jarptica [38.1K]

Answer:

1st,4th,5th option.

Step-by-step explanation:

Let evaluate each option.

A segment bisector is a line segment, ray, and/or line that bisects a line into two congruent parts. LM splits JK and KH into congruent parts. The first option is correct.

A perpendicular bisector is a line segment,ray and/or line that intersects a line segment,ray at a right angle. We don't have a perpendicular angle here and so that isn't a option.

M isn't a vertex of congruent angles as there is none in this figure.

The fourth and fifth option are correct. They both are on the segment bisector so they split the figure segments into two congruent parts. Since they are on the line segment and bisects it, they are considered the midpoint or middle point of the figure side.

7 0

3 years ago

PLEASE HELP !! ILL GIVE BRAINLIEST *EXTRA POINTS*.. <br> IM GIVING 40 POINTS !! DONT SKIP :((.

Liula [17]

Answer:

C

Step-by-step explanation:

Evolution is where organisms have developed over time from simple to complex. Therefore we can rule out D because older rock layers should have simple organisms and A because new rock layers should have more complex organisms. The previous organisms will be unlike those of today, therefore the correct answer is C.

7 0

3 years ago

If a represents the hypotenuse of this right triangle, which equation could you use to find the value of a?

Nitella [24]

The answer is D because they are both legs and you are looking for the hypotenuse

4 0

2 years ago

Read 2 more answers

Which best describes the vertex of the graph?

nydimaria [60]

Answer: C

Step-by-step explanation: