8 1 Volumes 8 2 Arc Length and

本章大綱 8. 1 Volumes 體積 8. 2 Arc Length and Area of a Surface of Revolution 曲線段的長度及旋轉體的表面積 8. 3 Applications to Physics and Engineering

8. 1 Volumes 體積

8. 1. 1 Cross Section Method 橫切面法

例題1. Show that the volume of a sphere of radius 解： is .

8. 1. 2 Solids of Revolution. The Disk Method圓盤法

例題3. Find its volume of the solid generated by revolving the region between the curves about the -axis. 解：

例題4. Find its volume of the solid generated by revolving the region bounded by about the line 解： .

8. 1. 3 The Washer Method

例題5. The region bounded by and is revolved about the to generate a solid. Find the volume of the solid. 解： 技巧：畫出區域R，並畫一線段橫越R且與旋轉軸垂直，找出橫越線段 的範 圍，即內外旋轉體軸半徑。

例題6. The region bounded by and is revolved about the to generate a solid. Find the volume of the solid. 解：

8. 1. 4 Cylindrical Shell Method 圓柱殼法

例題7. Find the volume of the solid generated by revolving the region about the y-axis. 解：

例題8. Find the volume of the solid generated by revolving the region about the x-axis 解：

8. 2 Arc Length and Area of a Surface of Revolution 曲線段的長度及旋轉體的表面積 8. 2. 1 Arc Length曲線段長

例題1. Find the length of the arc of the parabola 解： from .

例題2. Find the circumference of the circle 解： .

8. 2. 2 Length of a Parametric Curve

例題3. Find the length of the asteroid 解： .

8. 2. 3 Area of a Surface of Revolution 旋轉體的側表面積 說明：

例題4. Find the area of the surface of a sphere with radius 解： .

例題5. Find the area of the surface generated by revolving the curve about the 解： .

8. 2. 4 Surface Area of Revolution for a Parametric Curve

例題6. Find the area of the surface swept out by revolving the circle about the 解： .

8. 3 Applications to Physics and Engineering 8. 3. 1 work 功

8. 3. 2 Fluid Pressure液壓