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sergiy2304 [10]
2 years ago
8

Let C be the positively oriented square with vertices (0,0), (1,0), (1,1), (0,1). Use Green's Theorem to evaluate the line integ

ral ∫C7y2xdx+8x2ydy.
Mathematics
1 answer:
liq [111]2 years ago
4 0

Answer:

1/2

Step-by-step explanation:

The interior of the square is the region D = { (x,y) : 0 ≤ x,y ≤1 }. We call L(x,y) = 7y²x, M(x,y) = 8x²y. Since C is positively oriented, Green Theorem states that

\int\limits_C {L(x,y)} \, dx + {M(x,y)} \, dy = \int\limits^1_0\int\limits^1_0 {(Mx - Ly)} \, dxdy

Lets calculate the partial derivates of M and L, Mx and Ly. They can be computed by taking the derivate of the respective value, treating the other variable as a constant.

  • Mx(x,y) = d/dx 8x²y = 16xy
  • Ly(x,y) = d/dy 7y²x = 14xy

Thus, Mx(x,y) - Ly(x,y) = 2xy, and therefore, the line ntegral is equal to the double integral

\int\limits^1_0\int\limits^1_0 {2xy} \, dxdy

We can compute the double integral by applying the Barrow's Rule, a primitive of 2xy under the variable x is x²y, thus the double integral can be computed as follows

\int\limits^1_0\int\limits^1_0 {2xy} \, dxdy = \int\limits^1_0 {x^2y} |^1_0 \,dy = \int\limits^1_0 {y} \, dy = \frac{y^2}{2} \, |^1_0 = 1/2

We conclude that the line integral is 1/2

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Answer:

31 cm

Step-by-step explanation:

The complete question is attached.

A triangle is a polygon with three sides (three edges and three vertices). There are different types of triangles which are equilateral triangles, right triangles, scalene triangles, obtuse triangles, acute triangles, and isosceles triangles.

An equilateral triangle is a triangle in which all its sides are equal and all its angles are equal to 60°.

From the triangle, AB = BC = AC = 9.5 cm

Since the triangle is enlarged 3.25 times, hence:

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New length of BC = 31 cm to nearest cm

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3 years ago
The length of a rectangle is 2 cm more than four times the width. If the perimeter of the rectangle is 84 cm, what are its dimen
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Answer:

Length 34 cm

Width 8 cm

Step-by-step explanation:

Let

x-----> the length of the rectangle

y----> the width of the rectangle

we know that

The perimeter of rectangle is equal to

P=2(x+y)

P=84 cm

so

84=2(x+y)

42=x+y ----> equation A

x=4y+2 -----> equation B

substitute equation B in equation A and solve for y

42=(4y+2)+y

5y=42-2

y=40/5=8 cm

Find the value of x

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The dimensions are

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Width 8 cm

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What set of reflections would carry hexagon ABCDEF onto itself?
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When a set of reflections that carry a shape onto itself, it means that the final position of the shape will be the same as its original location

Reflections <em>(a) y=x, x-axis, y=x, y-axis </em>would carry the hexagon onto itself

First; we test the given options, until we get the true option

<u>(a) y=x, x-axis, y=x, y-axis</u>

The rule of reflection y =x is:

(x,y) \to (y,x)

The rule of reflection across the x-axis is:

(x,y) \to (x,-y)

So, we have:

(y,x) \to (y,-x)

The rule of reflection y =x is:

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So, we have:

(y,-x) \to (-x,y)

Lastly, the reflection across the y-axis is:

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So, we have:

(-x,y) \to (x,y)

So, the overall transformation is:

(x,y) \to (x,y)

Notice that, the original and final coordinates are the same.

This means that:

Reflections <em>(a) y=x, x-axis, y=x, y-axis </em>would carry the hexagon onto itself

Read more about reflections at:

brainly.com/question/938117

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