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Volgvan
2 years ago
7

Consider F and C below. F(x, y) = x4y5i + x5y4j, C: r(t) = t3 − 2t, t3 + 2t , 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f.

f(x, y) = (b) Use part (a) to evaluate C F · dr along the given curve C.
Mathematics
1 answer:
vovikov84 [41]2 years ago
5 0
\mathbf f(x,y)=x^4y^5\,\mathbf i+x^5y^4\,\mathbf j

We want a scalar function f(x,y) whose gradient is equivalent to the vector field. That means

\dfrac{\partial f}{\partial x}=x^4y^5\implies f(x,y)=\dfrac{x^5y^5}5+g(y)
\dfrac{\partial f}{\partial y}=x^5y^4\implies x^5y^4=x^5y^4+\dfrac{\mathrm dg}{\mathrm dy}
\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C

So (a)

f(x,y)=\dfrac{x^5y^5}5+C

which means (b)

\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(\mathbf r(1))-f(\mathbf r(0))=f(-1,3)-f(0,0)=-\dfrac{243}5-0=-\dfrac{243}5
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son4ous [18]

Answer:

75.44 Square Inches

Step-by-step explanation:

The diagram of the problem is produced and attached.

To determine the area of the cleaned sector:

Let the radius of the larger sector be R

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Area of the larger sector =\frac{\theta}{360}X\pi R^2

Area of the smaller sector =\frac{\theta}{360}X\pi r^2

Area of shaded part =Area of the larger sector-Area of the smaller sector

=\frac{\theta}{360}X\pi R^2-\frac{\theta}{360}X\pi r^2\\=\frac{\theta \pi}{360}X (R^2- r^2)

From the diagram, R=10 Inch, r=10-7=3 Inch, \theta=95^\circ

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7 0
3 years ago
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Hoochie [10]

Answer:

f(x-3)= x^2 -11x +24

Step-by-step explanation:

We want to find an equivalent expression for f(x-3)

To do this, we can simply substitute every 'x' variable in f(x) by 'x-3'

Therefore f(x-3)= (x-3)^2 -5*(x-3)

Lets re arrange the expression by grouping terms.

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3 0
3 years ago
If two planes intersect they will form a
Alja [10]

Answer:

They meet a line. Therefore, if two planes cut one another, then their intersection is a straight line.

Step-by-step explanation:


8 0
2 years ago
The shown bellow question answer
Serhud [2]

Answer:

The area of the shape can be divided into the area of the rectangle, and the area of the semi-circle.

The area of the rectangle can be found by 11*4=44

The area of a semi-circle can be found with the formula \frac{1}{2} \pi r^2 where r is the radius.

Since we know the diameter of the semi-circle is 4,

the radius will be 4 ÷ 2 = 2.

Therefore, the area of the semi-circle is \frac{1}{2}\pi 2^2=\frac{1}{2}\pi*4=2\pi

Therefore, the area of the shape is 44+2\pi or 50.283cm^2 (3 decimal places)

3 0
2 years ago
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tatuchka [14]

Answer:

It is p = 1

Step-by-step explanation:

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6 0
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