Q. 63 - Calculus

Question

We may express the volume of a right circular cylinder by using the function of two variables, $V(r,h)=\pi r^{2}h$, where r is the radius of either end of the cylinder and h is the height of the cylinder. What is the domain of the function V(r, h)? Express the surface area, S, of a cylinder as a function of variables r and h. 

Step-by-Step Solution

Answer

The domain of the function is $\left\{\right(x,y)/x\in r,y\in h,x,y\ge 0\}$

The surface area of the cylinder can be given as $2\pi r(h+r)$.

Step 1: Given information

We are given the volume of cylinder as $V(r,h)=\pi r^{2}h$.

Step 2: Find the domain of the function

As the radius and height is always positive the domain of the function is $\left\{\right(x,y)/x\in r,y\in h,x,y\ge 0\}$

Step 3: Find the surface area

The surface area of a cylinder is the sum of Curved Surface +and Area of Circular bases 

Hence we have

Total surface area= $$\begin{gathered} 2\pi rh+2\pi r^2 \\ =2\pi r(h+r) \end{gathered}$$

Hence we expressed the surface area of cylinder as a function of r and h