Respuesta :
The surface area of the torus with radius 10, which can be represented parametrically is 40π²
Torus and its area can be defined parametrically as:
r(θ,ϕ)= ((10+cosϕ)cosθ, (10+cosϕ)sinθ, sinϕ), 0≤θ≤2π, 0≤ϕ≤2π.
First, we need to calculate the partial derivatives of
r:rθ=(−(10+cosϕ)sinθ,(10+cosϕ)cosθ,0)
rϕ=(−sinϕcosθ,−sinϕsinθ,cosϕ)
rθ×rϕ= ∣ i j k|
|−(10+cosϕ)sinθ (10+cosϕ)cosθ 0|
|−sinϕcosθ −sinϕsinθ cosϕ∣
=(10+cosϕ) (cosθcosϕi+ sinθcosϕ j+ cosϕk)
thus,
||(10+cosϕ)(cosθcosϕi+sinθcosϕj+cosϕk)||
=(10+cosϕ) √(cosθcosϕ)²+(sinθcosϕ)²+(cosϕ)²
=(10+cosϕ)√cos²θcos²ϕ+ sin²θcos²ϕ +sin²ϕ
=(10+cosϕ)√cos²ϕ+sin²ϕ
=10+cosϕ
the surface integral is given as:
S=∫∫(10+cosϕ) dϕ dθ , [0,2π]
=∫[10ϕ+sinϕ] [0,2π] dθ, [0,2π]
=∫20πdθ, [0,2π]
=(20πθ) [0,2π]
=40π²
Therefore, 40π² is the surface area represented parametrically.
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