Green's theorem says the circulation of $\vec F$ along the rectangle's border $C$ is equal to the integral of the curl of $\vec F$ over the rectangle's interior $D$ .

Given $\vec F(x,y)=2xy\,\vec\imath$ , its curl is the determinant

$\det\begin{bmatrix}\frac\partial{\partial x}&\frac\partial{\partial y}\\2xy&0\end{bmatrix}=\dfrac{\partial(0)}{\partial x}-\dfrac{\partial(2xy)}{\partial y}=-2x$

So we have

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D-2x\,\mathrm dx\,\mathrm dy=-2\int_0^3\int_0^8x\,\mathrm dx\,\mathrm dy=\boxed{-192}$

6 0

3 years ago

I need help answering this ASAP

Leokris [45]

Answer:

x greater than and equal to -7

Step-by-step explanation:

Parabola touches at -7 and is going straight towards all positive integers

Therefore the domain is all numbers greater than or equal to -7

8 0

2 years ago

Is 15 kilometers equivalent to 1,500,000 centimeters

Ann [662]

Hi
Yes 15 kilometers is 1500000 cm

This is because in metric conversion, you multiply by ten each time you convert to the next unit.

Heres a useful way to remember

K ing (kilo)
H enry (hecto)
D oesn't (deca)
U sually (unit) <-metre, litre ect.
D rink ( deci)
C hocolate (centi)
M ilk (milli)

6 0

3 years ago