Definite Integrals: Application to Paragliding

Paragliders are like modern parachutes that are designed for best gliding properties. Therefore they are not used from airplanes but pilots launch from hills or mountain slopes. In good weather conditions they can stay aloft for hours and can travel for ore than hundred miles. They have a single wing that is inflated when air enters through openings in the front ("ram-air" system).

The size of the wing determines how much weight the wing can cary. To be more specific: The "projected area" of the wing that means the area that is horizontal and therefore providing "lift". In this example we want to calculate the area of a paraglider wing from its (projected) wingspan and its "aspect ratio".

The above drawing shows the projected area of a modern paragliding wing. Its technical data are given in the table below.

Determine the Projected Area of the Wing:

The equations for each of the semi ellipses can be derived by solving the defining equation:

for y:

 

Problems:

• Integrate these functions to determine the area between these curves.
• Measure a set of (x,y)-coordinates of the wing from the drawing and numerically integrate the obtained curves.

Notes: 

• Graphics 1-2 and photo courtesy NOVA Paragliding
• Solutions can be found here.