How to Find the Area Enclosed by The "Boat Propeller" Curves

Added: 2026-05-16

[00:00:00]This is the boat propeller curve. It's equation looks intimidating, but symmetry is the key to unlocking it.

[00:00:07]Let's find the area it encloses. First, we determine where the curve actually exists. Since radius squared must be positive, the graph appears only on four specific intervals. So, we find the area of one lobe, then multiply by four.

[00:00:22]Next, we set up the polar area integral.

[00:00:24]Using a basic trigonometric identity, we rewrite the denominator into a form perfect for substitution. Let you equal the square root of 3 * cosine of 4 theta.

[00:00:35]As we switch variables, watch how the limits transform into the U domain. Now, the integral becomes a classic arc tangent form.

[00:00:43]Evaluating from negative root 3 to positive root 3 gives 2 pi over 3.

[00:00:49]Multiply by the leading coefficient, and the final area is 4 pi root 3 over 9.

[00:00:54]Complex curve, elegant solution. Follow for the next integral in the series.