# Early Transcendental Functions (Smith-Minton), 3rd Edition

## Chapter 13: Multiple Integrals

 (21.0K) The design of modern sports equipment has become a sophisticated engineering enterprise. Many innovations can be traced back to a brilliant engineer but mediocre athlete named Howard Head. As an aircraft engineer in the 1940s, Head became frustrated learning to ski on the wooden skis of the day. Following years of experimentation, Head revolutionized the ski industry by introducing metal skis designed using principles borrowed from aircraft design. By 1970, Head had retired from the Head Ski Company as a wealthy ski mogul. He quickly became frustrated by his slow progress learning to play tennis, a sport then played exclusively with wooden rackets. Head again focused on his equipment, reasoning that a larger racket would twist less and therefore be easier to control. However, years of experimentation showed that large wooden rackets either broke easily or were too heavy to swing. Given that Head's metal skis were successful largely because they reduced the twisting of the skis in turns, it is not surprising that his experimentation turned to oversized metal tennis rackets. The rackets that Head eventually marketed as Prince rackets revolutionized tennis racket design. As the accompanying diagram shows, the sweet spot of the oversized racket is considerably larger than the sweet spot of the smaller wooden racket. In this chapter, we introduce double and triple integrals, which are needed to compute the mass, moment of inertia and other important properties of three-dimensional solids. The moment of inertia is a measure of the resistance of an object to rotation. As shown in the exercises in section 13.2, compared to smaller rackets, the larger Head rackets have a larger moment of inertia and thus, twist less on off-center shots. Engineers use similar calculations as they test new materials for strength and weight for the next generation of sports equipment.