Chap8_Sec2 - 8 FURTHER APPLICATIONS OF INTEGRATION OF...

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FURTHER APPLICATIONS FURTHER APPLICATIONS OF INTEGRATION OF INTEGRATION 8
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8.2 Area of a Surface of Revolution In this section, we will learn about: The area of a surface curved out by a revolving arc. FURTHER APPLICATIONS OF INTEGRATION
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SURFACE OF REVOLUTION A surface of revolution is formed when a curve is rotated about a line. Such a surface is the lateral boundary of a solid of revolution of the type discussed in Sections 6.2 and 6.3
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AREA OF A SURFACE OF REVOLUTION We want to define the area of a surface of revolution in such a way that it corresponds to our intuition. If the surface area is A , we can imagine that painting the surface would require the same amount of paint as does a flat region with area A .
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Let’s start with some simple surfaces. AREA OF A SURFACE OF REVOLUTION
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CIRCULAR CYLINDERS The lateral surface area of a circular cylinder with radius r and height h is taken to be: A = 2 πrh We can imagine cutting the cylinder and unrolling it to obtain a rectangle with dimensions of 2 πrh and h .
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CIRCULAR CONES We can take a circular cone with base radius r and slant height l , cut it along the dashed line as shown, and flatten it to form a sector of a circle with radius and central angle θ = 2 πr / l .
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CIRCULAR CONES We know that, in general, the area of a sector of a circle with radius l and angle θ is ½ l 2 θ .
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CIRCULAR CONES So, the area is: Thus, we define the lateral surface area of a cone to be A = πr l. 2 2 1 1 2 2 2 r A l l rl l π θ = = = ÷
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AREA OF A SURFACE OF REVOLUTION What about more complicated surfaces of revolution?
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AREA OF A SURFACE OF REVOLUTION If we follow the strategy we used with arc length, we can approximate the original curve by a polygon. When this is rotated about an axis, it creates
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Chap8_Sec2 - 8 FURTHER APPLICATIONS OF INTEGRATION OF...

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