Volume: The Shell MethodSlide 2Slide 3Shell MethodThe ShellTry It Out!Hints for Shell MethodRotation About x-AxisRotation About Non-coordinate AxisSlide 10Try It OutSlide 12AssignmentVolume: The Shell MethodLesson 7.3 2410 169y x x Find the volume generated when this shape is revolved about the y axis.We can’t solve for x, so we can’t use a horizontal slice directly. 2410 169y x x If we take a vertical sliceand revolve it about the y-axiswe get a cylinder.Shell Method•Based on finding volume of cylindrical shellsAdd these volumes to get the total volume•Dimensions of the shellRadius of the shellThickness of the shellHeightThe Shell•Consider the shell as one of many of a solid of revolution•The volume of the solid made of the sum of the shellsf(x)g(x)xf(x) – g(x)dx[ ]2 ( ) ( )baV x f x g x dxp= -�Try It Out!•Consider the region bounded by x = 0, y = 0, and 28y x= -2 2202 8V x x dxp= � -�Hints for Shell Method•Sketch the graph over the limits of integration•Draw a typical shell parallel to the axis of revolution•Determine radius, height, thickness of shell•Volume of typical shell•Use integration formula 2 radius height thicknessp�� � �2baVolume radius height thicknessp= � � ��Rotation About x-Axis•Rotate the region bounded by y = 4x and y = x2 about the x-axis•What are the dimensions needed?radiusheightthicknessradius = yheight = 4yy -thickness = dy16024yV y y dyp� �= � -� �� ��Rotation About Non-coordinate Axis•Possible to rotate a region around any line•Rely on the basic concept behind the shell methodx = af(x)g(x)2sV radius height t hi cknessp= �� � �Rotation About Non-coordinate Axis•What is the radius?•What is the height?•What are the limits?•The integral:x = af(x)g(x)a – xf(x) – g(x)x = crc < x < a[ ]( ) ( ) ( )acV a x f x g x dx= - -�Try It Out•Rotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2•Determine radius, height, limits4 – x2 4 – x2 r = 2 - xr = 2 - x20�Try It Out•Integral for the volume is2202 (2 ) (4 )V x x dxp= - � -�Assignment•Lesson 7.3•Page 472•Exercises 1 – 25 odd•Lesson 7.3B•Page 472•Exercises 27, 29, 35, 37, 41, 43,
View Full Document
Unlocking...