Calculus: Understanding Change: Unit 2: Built by Calculus: Dam Construction

How do doctors and pharmacologists use calculus to prescribe drugs? How do engineers use area, volume, and pressure equations to design dams? Calculus: Understanding Change gives students a powerful demonstration of the critical role calculus plays in our lives.

Unit 2: Built by Calculus: Dam Construction (11:57)

Summary of the Video

Pastoral aquatic scenes soon change to violent ones in the opening of this video. As Johnny Cash sings, we watch the infamous Johnstown flood that was caused by the collapse of a poorly designed dam. This introduction emphasizes the importance of building structures to last and the role of the engineer in doing so. We hear that the engineer must apply calculus in order to build a quality dam. In particular, the areas of the dam's edges, the dam's volume and the pressure of the water against its gates are all important in dam design.

A computer-generated model of the dam allows us to view the dam from its edge and top. We can see that the dam bows against the water through an angle of 150°, which creates a very stable structure. Through scenes of blasting and construction, we see the surfaces of the surrounding rock being prepared to accept the edges of the dam and are introduced to our first problem: What is the area of the dam's edge?

The dam's edge is divided into three sections: a rectangle, a triangle, and the area under a parabola. The areas of the first two are found by the standard geometric methods. The area under the parabola is first divided into four rectangles, whose total area falls short of the area we want. More rectangles are used, and we can see that as the width of the rectangle becomes smaller, the area approaches a limit. The appropriate definite integral is applied to determine the precise area under the parabola.

Scenes of concrete construction remind us of the engineer's second problem: the amount of concrete the dam requires. We see that the dam's volume can be thought of as a slice rotated through a 150° angle. l11is slice is divided into rectangles and each is rotated to form a "slab." Just as rectangles can be used to approximate the area under a curve, the slabs can be used to approximate the volume of the dam. We find the volume of one slab by multiplying its length, width, and height, then adding the volumes of the individual slabs to approximate the dam's volume. Once again, by letting the width of the individual slabs get smaller, we approach a limit, and apply the appropriate integral to obtain the precise volume of the dam.

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