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dedylja [7]
2 years ago
6

Use Green's Theorem to evaluate F · dr. C (Check the orientation of the curve before applying the theorem.) F(x, y) = e−x + y2,

e−y + x2 , C consists of the arc of the curve y = cos(x) from − π 2 , 0 to π 2 , 0 and the line segment from π 2 , 0 to − π 2 , 0
Mathematics
1 answer:
Yuki888 [10]2 years ago
3 0

Notice that C is traversed clockwise. Green's theorem applies to curves with a counterclockwise orientation, so we'll have to multiply the area integral by -1.

By Green's theorem, with the vector field \vec F(x,y)=P(x,y)\,\vec\imath+Q(x,y)\,\vec\jmath,

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=-\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,\mathrm dA

where D is the region with boundary C. The partial derivatives are

\dfrac{\partial(e^{-y}+x^2)}{\partial x}=2x

\dfrac{\partial(e^{-x}+y^2)}{\partial y}=2y

so that the double integral is

\displaystyle\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,\mathrm dA=\int_{-\pi/2}^{\pi/2}\int_0^{\cos x}2(y-x)\,\mathrm dy\,\mathrm dx=\boxed{\frac\pi2}

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Answer:

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Step-by-step explanation:

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lyudmila [28]

Answer:

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Step-by-step explanation:

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Which of the following sets would have a graph with an open circle on 2 and a ray pointing left on the number line?
MaRussiya [10]
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3 years ago
Solve the equation. Check for extraneous solutions. |x - 4| = 5x + 12<br><br> x = []
kykrilka [37]

The solution of the equation is x = -4/3.

<h3>What does it mean to solve an equation?</h3>

An equation represents equality of two or more mathematical expression.

Solutions to an equation are those values of the variables involved in that equation for which the equation is true.

WE have been given an equation as;

|x - 4| = 5x + 12

In an absolute value equation, we solve the original expression as our first equation. Our second one is that we multiply the right side by -1.

Case 1: original equation

|x - 4| = 5x + 12

x - 4 = 5x + 12

x - 5x = 12 + 4

-4x = 16

x = -4

Case 2: Opposite equation

|x - 4| = 5x + 12

x - 4 = - (5x + 12)

x - 4 = - 5x - 12

x + 5x = -12 + 4

6x = -8

x = -4/3

Now we have two solutions. We need to check for extraneous solutions because of all the manipulations;

Check:

|x - 4| = 5x + 12

use x = -4

|-4 - 4| = 5(-4) + 12

| -8 | = -20 + 12

8 = -8              

Thus, it is Not a solution

Now,  |x - 4| = 5x + 12

use x =  -4/3

| -4/3 - 4| = 5( -4/3) + 12

|-16/3 | = -20/3 + 12

|-16/3 | = 16/3

16/3 = 16/3

Thus, it is the  Solution.

Learn more about solving equations here:

brainly.com/question/18015090

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5 0
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slavikrds [6]

Answer:

The answer will be 1/9

Step-by-step explanation:

any non zero expression will always be 1. you now have evaluate the powe (-3)^2 which will be 9. Answer should be 1/9. Good Luck!

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