Q 1. - 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.”

Find the volume of the cylinder.

Step-by-Step Solution

Answer

Volume is 5 square units.

Step 1. Given information

Cardioid is $r=1+\cos \theta$

Step 2. Explanation

Consider the equation of cardioid as $r=1+\cos \theta$

Volume of a cylinder $=\pi r^{2}h$ where $r,h$ denote radius, height respectively.

Interior of the cardioid is translated one unit perpendicular to the xy-plane to define a cylinder.

Volume width="116" style="max-width: none; vertical-align: -10px;" $=\int 1+\cos \theta d\theta$

               $=5$ cubic units