Mathematics, 03.07.2020 22:01 sanji8442

Use Green's Theorem to evaluate ?C F·dr. (Check the orientation of the curve before applying the theorem.) F(x, y) =< x + 4y3, 4x2 + y>

C consists of the arc of the curve y = sin x from (0, 0) to (p, 0) and the line segment from (p, 0) to (0, 0).

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Answer from: vildhjap42ges

Step-by-step explanation:

given a field of the form F = (P(x,y),Q(x,y) and a simple closed curve positively oriented, then

$\int_{C} F \cdot dr = \int_A \frac{dQ}{dx} - \frac{dP}{dy} dA$ where A is the area of the region enclosed by C.

In this case, by the description we can assume that C starts at (0,0). Then it goes the point (pi,0) on the path giben by y = sin(x) and then return to (0,0) along the straigth line that connects both points. Note that in this way, the interior the region enclosed by C is always on the right side of the point. This means that the curve is negatively oriented. Consider the path C' given by going from (0,0) to (pi,0) in a straight line and the going from (pi,0) to (0,0) over the curve y = sin(x). This path is positively oriented and we have that

$\int_{C} F\cdot dr = - \int_{C'} F\cdot dr$

We use the green theorem applied to the path C'. Taking P = x+4y^3, Q = 4x^2+y we get

$\int_{C'} F\cdot dr = \int_{A} 8x-12y^2dA$

A is the region enclosed by the curves y =sin(x) and the x axis between the points (0,0) and (pi,0). So, we can describe this region as follows

$0\leq x \leq \pi, 0\leq y \leq \sin(x)$

This gives use the integral

$\int_{A} 8x-12y^2dA = \int_{0}^{\pi}\int_{0}^{\sin(x)} 8x-12y^2 dydx$

Integrating accordingly, we get that $\int_{C'} F\cdot dr = 8\pi - \frac{16}{3}$

$\int_{C} F cdot dr = - (8\pi - \frac{16}{3}) = \frac{16}{3} - 8\pi$

Answer from: Quest

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

= 2×(-2)+(-8)×(6)+(-8)×(-5)

= -4-48+40

= -12

hence, v.w= -12

Answer from: Quest

a is 8.b is 10.c is 10.5. d is -.5

step-by-step explanation:

just minus the smallest number from the biggest.

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