Question

use green's theorem for area to find the total grazing area of the goat

Answer 1

Answer:

Step-by-step explanation:

Try to do a Riemann sum. For 0<θ<π/2, consider the strip of land covered as the contact angle moves from θ to θ+dθ. Integrate that. Then, for π/2<θ<π, the end of the rope is coming back west, and the strip of land you measure is the inaccessible part. Integrate that and subtract from the first answer.

When the contact angle is θ, a bound on the region is

P1(−16cosθ+16(π−θ)sinθ,16sinθ+16(π−θ)cosθ)

Another bound is on the south side,

P2(−16cosθ+16(π−θ)sinθ,−16sinθ−16(π−θ)cosθ)

Then, allowing just a bit more contact on the barn, to θ+dθ, you get

P3(−16cos(θ+dθ)+16(π−(θ+dθ))sin(θ+dθ),16sin(θ+dθ)+16(π−(θ+dθ))cos(θ+dθ))

and similar for P4. There is a region P1P2P4P3 which is a trapezoid, whose height is P1y and whose width is dθ times dP1x/dθ