3 years ago

The perimeter of Macy's rectangular room is 40 feet. Two sides of the room measure 8 feet. How long are the other two sides?

1 answer:

7 0

The side oppoiste to the 8 ft side must also = 8 feet
which is a total of 16 feet

so the other 2 side measure (40-16) / 2 = 12 feet

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C xy2 ds, c is the right half of the circle x2 y2 = 4 oriented counterclockwise

Kryger [21]

Parameterize the contour by

$C:\mathbf r(t)=(2\cos t,2\sin t)\implies \dfrac{\mathrm d\mathbf r}{\mathrm dt}=(-2\sin t,2\cos t)$
$\implies\mathrm dS=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt=2$

where $-\dfrac\pi2\le t\le\dfrac\pi2$ . Then the line integral is given by

$\displaystyle\int_C xy^2\,\mathrm dS=2\int_{-\pi/2}^{\pi/2}(2\cos t)(2\sin t)^2\,\mathrm dt$
$=\displaystyle16\int_{-\pi/2}^{\pi/2}\cos t\sin^2t\,\mathrm dt=\dfrac{32}3$