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leonid [27]
3 years ago
12

Use the Fundamental Theorem for Line Integrals to find Z C y cos(xy)dx + (x cos(xy) − zeyz)dy − yeyzdz, where C is the curve giv

en by ~r(t) = (t, πt/2, 1 − t), 0 ≤ t ≤ 1.
Mathematics
1 answer:
Harrizon [31]3 years ago
3 0

Answer:

The Line integral is π/2.

Step-by-step explanation:

We have to find a funtion f such that its gradient is (ycos(xy), x(cos(xy)-ze^(yz), -ye^(yz)). In other words:

f_x = ycos(xy)

f_y = xcos(xy) - ze^{yz}

f_z = -ye^{yz}

we can find the value of f using integration over each separate, variable. For example, if we integrate ycos(x,y) over the x variable (assuming y and z as constants), we should obtain any function like f plus a function h(y,z). We will use the substitution method. We call u(x) = xy. The derivate of u (in respect to x) is y, hence

\int{ycos(xy)} \, dx = \int cos(u) \, du = sen(u) + C = sen(xy) + C(y,z)  

(Remember that c is treated like a constant just for the x-variable).

This means that f(x,y,z) = sen(x,y)+C(y,z). The derivate of f respect to the y-variable is xcos(xy) + d/dy (C(y,z)) = xcos(x,y) - ye^{yz}. Then, the derivate of C respect to y is -ze^{yz}. To obtain C, we can integrate that expression over the y-variable using again the substitution method, this time calling u(y) = yz, and du = zdy.

\int {-ye^{yz}} \, dy = \int {-e^{u} \, dy} = -e^u +K = -e^{yz} + K(z)

Where, again, the constant of integration depends on Z.

As a result,

f(x,y,z) = cos(xy) - e^{yz} + K(z)

if we derivate f over z, we obtain

f_z(x,y,z) = -ye^{yz} + d/dz K(z)

That should be equal to -ye^(yz), hence the derivate of K(z) is 0 and, as a consecuence, K can be any constant. We can take K = 0. We obtain, therefore, that f(x,y,z) = cos(xy) - e^(yz)

The endpoints of the curve are r(0) = (0,0,1) and r(1) = (1,π/2,0). FOr the Fundamental Theorem for Line integrals, the integral of the gradient of f over C is f(c(1)) - f(c(0)) = f((0,0,1)) - f((1,π/2,0)) = (cos(0)-0e^(0))-(cos(π/2)-π/2e⁰) = 0-(-π/2) = π/2.

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Step-by-step explanation:

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Step-by-step explanation:

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The length of a rectangle is 5 ft less than three times the width, and the area of the rectangle is 50 ft². Find the dimensions
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Answer

Length = 10 ft

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Explanation

Area of the rectangle given = 50 ft²

Let the width of the rectangle be x

So this means the length of the rectangle will be 3x - 5

What to find:

The dimensions of the rectangle.

Step-by-step solution:

Area of a rectangle = length x width

i.e A = L x W

Put A = 50, L = 3x - 5, W = x into the formula.

\begin{gathered} 50=(3x-5)x \\ 50=3x^2-5x \\ 3x^2-5x-50=0 \end{gathered}

The quadratic equation can now be solve using factorization method:

\begin{gathered} 3x^2-5x-50=0 \\ 3x^2-15x+10x-50=0 \\ 3x(x-5)+10(x-5)=0 \\ (3x+10)(x-5)=0 \\ 3x+10=0\text{ }or\text{ }x-5=0 \\ 3x=-10\text{ }or\text{ }x=5 \\ x=-\frac{10}{3}\text{ }or\text{ }x=5 \end{gathered}

Since the dimension can not be negative, hence the value of x will be = 5.

Therefore, the dimensions of the rectangle will be:

\begin{gathered} Length=3x-5=3(5)-5=15-5=10\text{ }ft \\  \\ Width=x=5\text{ }ft \end{gathered}

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To complete this component you will need to insert parentheses in Expression 2 to change the order of operations so the expressi
Alika [10]

Given:

The expression is 5+4\times 2+6-2\times 2-1.

To find:

The position of parentheses inserted in the expression to get the value 19

Step-by-step explanation:

We have,

5+4\times 2+6-2\times 2-1

Now,

(5+4)\times 2+6-2\times 2-1=9\times 2+6-4-1

(5+4)\times 2+6-2\times 2-1=18+6-5

(5+4)\times 2+6-2\times 2-1=18+1

(5+4)\times 2+6-2\times 2-1=19

There are more ways to apply the parenthesis, but we do not get 19.

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And many more possibilities.

Therefore, the expression after inserting the parentheses is (5+4)\times 2+6-2\times 2-1.

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