Answer to Question #98240 in Calculus for Mimo

Question

Show that the surface Area of Cylinder = 2πRH
(cylindrical coordinate system)

Answer 1

For calculating the curved surface of this geometrical figure, we need to first wrap a cylindrical can with a sheet of paper. The paper should be wrapped in such a manner with tape that it fits accurately with this cylindrical can.

Now, remove the paper and cut this cylindrical paper, parallel to its axis, making it rectangular in shape. This activity gives us rectangle having $length = 2.π.r$ and $breadth = h.$

$Surface area of cylinder = area of the rectangle = 2.π.r.h$

(Answer)

Proof using cylindrical coordinate system

let the cylinder represented in cylindrical coordinat es (r,θ,z)

(r,θ,z) where r is the radius from the z-axis, θ is the azimuthal angle. Now the surface area of a small element of the cylinder will be given by $dA=rdθdz$.We seek to integrate around the cylinder $0≤θ≤2π$ and $0≤z≤h$

with a fixed radius r. The area of the cylinder is then the integral of dA,

$Area of cylinder = \int{dA}= \iint{rd(\theta)dz}$

$\theta$ ranges from 0 to 2π.z ranges from 0 to h.

$Area of cylinder = A=2.π.r. \int{dz}$

$Area of cylinder = A= 2πrh$

(Answer)