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lys-0071 [83]
3 years ago
15

Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise.

Mathematics
1 answer:
anzhelika [568]3 years ago
3 0

By Green's theorem, the circulation of \vec F(x,y)=\langle2xy^2,4x^3+y\rangle around C is equal to

\displaystyle\iint_D\frac{\partial(4x^3+y)}{\partial x}-\frac{\partial(2xy^2)}{\partial y}\,\mathrm dx\,\mathrm dy

where D is the region with C as its boundary. The integral is equivalent to

\displaystyle\int_0^\pi\int_0^{\sin x}(12x^2-4xy)\,\mathrm dy\,\mathrm dx=\boxed{\frac{23\pi^2}2-48}

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Solve the equation for all real values of x.<br> cosxtanx - 2 cos x=-1
IceJOKER [234]

Solution to equationcosxtanx - 2 cos^2 x=-1 for all real values of x is  x=2k\pi + \frac{\pi}{6}  , x=2k\pi + \frac{5\pi}{6} .

<u>Step-by-step explanation:</u>

Here we have , cosxtanx - 2 cos^2 x=-1. Let's solve :

⇒  cosxtanx - 2 cos^2 x=-1

⇒  cosx(\frac{sinx}{cosx}) - 2 cos^2 x=-1

⇒  sinx = 2 cos^2 x-1

⇒  sinx = 2 (1-sin^2x)-1

⇒  sinx = 1-2sin^2x

⇒  2sin^2x+sinx-1=0

By quadratic formula :

⇒ sinx = \frac{-b \pm \sqrt{b^2-4ac} }{2a}

⇒ sinx = \frac{-1 \pm \sqrt{1^2-4(2)(-1)} }{2(2)}

⇒ sinx = \frac{-1 \pm3}{4}

⇒ sinx = \frac{1}{2} , sinx =-1

⇒ sinx = sin\frac{\pi}{6} , sinx = sin\frac{3\pi}{2}

⇒ x=\frac{\pi}{6} , x=\frac{3\pi}{2}

But at x=\frac{3\pi}{2} we have equation undefined as cos\frac{3\pi}{2}=0 . Hence only solution is :

⇒ x=\frac{\pi}{6}

Since , sin(\pi -x)=sinx

⇒ x=\pi -\frac{\pi}{6} = \frac{5\pi}{6}

Now , General Solution is given by :

⇒ x=2k\pi + \frac{\pi}{6}  , x=2k\pi + \frac{5\pi}{6}

Therefore , Solution to equationcosxtanx - 2 cos^2 x=-1 for all real values of x is  x=2k\pi + \frac{\pi}{6}  , x=2k\pi + \frac{5\pi}{6} .

3 0
3 years ago
I need helpppp (10 points )
irga5000 [103]

Answer:  The flagpole is about 22.3 feet tall.

This is roughly the size of a two story building, assuming each floor is 10 feet or so.

Since this height is under 25 ft, this flagpole is in compliance with the regulation.

=====================================================

Work Shown:

The horizontal leg of the triangle is 36 ft. This is the adjacent leg to the reference angle 25 degrees.

The vertical leg of the triangle is x-5.5, and this leg is the opposite side to the reference angle 25 degrees.

Use the tangent rule to connect the opposite and adjacent sides.

Solve for x.

tan(angle) = opposite/adjacent

tan(25) = (x-5.5)/36

36*tan(25) = x-5.5

x-5.5 = 36*tan(25)

x = 36*tan(25)+5.5

x = 22.28707569358

x = 22.3 ft is the approximate height of the flagpole

Make sure your calculator is in degree mode.

The height is not over 25 ft, so this flagpole meets the requirements.

4 0
3 years ago
The sum of two numbers is 19 . When the second number is subtracted from the first​ number, the difference is 17 . Find the two
zhenek [66]

Answer:

19 and 2

Step-by-step explanation:

Let x represent the first number and y represents second number

x + y = 19

x - y = 17 add two equations up

2x = 36 divide both sides by 2

x = 19 and since the difference is 17 the other number is 2

4 0
2 years ago
The circumference of a circle is 24π in. What is the area, in square inches? Express your answer in terms of
inessss [21]

Answer:

it is 29in

Step-by-step explanation:

3 0
3 years ago
The ABC Toy Company is creating two similar pieces for board game, as shown below. Find the value of y that makes the two pieces
Ira Lisetskai [31]
A shape that is similar to another shape will be enlarged by a scale factor. 
Each corresponding sides should give the same scale factor

Side GI corresponds to side JL
Side GH corresponds to side JK
Side HI corresponds to side KL

The correspond sides whose length are given is 
GH = 4 and JK = 8

Side JK is twice longer than GH

All the other sides of triangle JKL are twice longer than triangle GHI, so we want that the side JL to be twice of the sides GI

Side GI = 6
Side JL = 6 × 2 = 12

Answer: y = 12


8 0
3 years ago
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