Final answer:
The surface area of the half-cylinder with radius 3 and height 8 can be found by multiplying the circumference of the half circle, πr, by the height, giving us an area of 24π. This is represented by the double integral ∫_0^π ∫_0^8 3 dz dθ.
Step-by-step explanation:
The student asked to find the area of the surface of the half cylinder with the given parametric description. Since we have a half cylinder, we can find its surface area by calculating the area of the curved surface without the circular ends.
The curved surface of a half cylinder can be thought of as a rectangle unrolled, with width equal to the height of the cylinder and length equal to the circumference of the half circle (which is πr for a half circle).
The surface area A of a half-cylinder can be found using the formula:
A = (Circumference of half circle) × (Height of cylinder)
A = (π × r) × h
Where r is the radius and h is the height. Inserting the given values, r = 3 and h = 8, we have:
A = (π × 3) × 8
A = 24π
Therefore, the integral for the surface area using the parameterization u = θ and v = z is:
∫∫ πθ dz dθ
The limits for u are from 0 to π, and for v from 0 to 8, so the double integral is:
∫_0^π ∫_0^8 3 dz dθ
Which simplifies to:
A = 24π