Parametric and Polar Integration. Area Enclosed Parametrically.
Find the area enclosed by the path of the Scrambler (a popular carnival ride) if its path is represented by
(Curve is traced out counterclockwise once for 0 t 2)
The equation for the length of a parametrized curve is similar to our previous “length of curve” equation:
(Notice the use of the Pythagorean Theorem.)
(proof on pg. 721)
Find the distance traveled by the point in one complete trip about the larger circle.
Likewise, the equations for the surface area of a parametrized curve are similar to our previous “surface area” equations:
The length of an arc (in a circle) is given by r.q when q is given in radians.
For a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula:
We can use this to find the area inside a polar graph.
Example: Find the area enclosed by:
Specifically a Cardiod)
To find the area between curves, subtract:
Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.
Example: Find the area inside and outside
To establish bounds, we must find where the two curves intersect.
Area of one leaf times 4:
Area of four leaves:
Again, for polar graphs:
If we find derivatives and plug them into the formula, we (eventually) get:
There is also a surface area equation similar to the others we are already familiar with:
When rotated about the x-axis: