Find the surface area of the helicoid (spiral

Answered question

2022-04-11

Find the surface area of the helicoid (spiral ramp) given by 𝑟⃗(𝑢, 𝑣) = 〈𝑢 cos 𝑣 , 𝑢 sin 𝑣 , 𝑣〉 for 0 ≤ 𝑢 ≤ 1 and 0 ≤ 𝑣 ≤ 𝜋. Use the following function to graph the helicoid in GeoGebra. surface(x,y,z, parameter variable 1, lower bound, upper bound, parameter variable 2, lower bound, upper bound)

Answer & Explanation

Vasquez

Vasquez

Expert2023-04-27Added 669 answers

The helicoid is given by the vector equation:
𝐫(u,v)=ucosv,usinv,v0u1,0vπ
To find its surface area, we need to compute the magnitude of the partial derivatives 𝐫u and 𝐫v and take their cross product:
𝐫u=cosv,sinv,0
𝐫v=usinv,ucosv,1
𝐫u×𝐫v=|𝐢𝐣𝐤cosvsinv0usinvucosv1|=usinv,ucosv,u
The magnitude of this vector is (usinv)2+(ucosv)2+u2=u2.
Therefore, the surface area of the helicoid is given by the integral:
A=010π𝐫u×𝐫vdudv=010πu2dudv
Evaluating this integral, we get:
A=20π01ududv=π2
Therefore, the surface area of the helicoid is π2.

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