Q 2. - Calculus

Question

The cardioid defined by r = 1 + cos θ and its interior is translated one unit perpendicular to the xy-plane to define a “cylinder.”

Surface area: Find the surface area of the cylinder.

Remember to include the top and bottom of the solid

in your calculations.

Step-by-Step Solution

Answer

Answer is 9 square units.

Step 1. Given information

The cardioid defined by r = 1 + cos θ and its interior is translated one unit perpendicular to the xy-plane to define a “cylinder.”

Step 2. Explanation

The cardioid defined by r = 1 + cos θ and its interior is translated one unit perpendicular to the xy-plane to define a “cylinder.”

Surface area of a cylinder $=2\mathrm{\pi r}\left(\mathrm{r}+\mathrm{h}\right)$ where $r,h$ denote radius, height of a cylinder respectively.

So, 

Surface area of a cylinder width="93" style="max-width: none; vertical-align: -10px;" $=\int 1+\cos \theta$

                                             $=9$ square units