160_6_3_Disk_Washer_6_4_ShellCOMPLETE.pdf - 6.3 Disk Method...

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6.3 Disk Method: intro A line segment to a single point can be rotated around the x -axis to create a disk. If each point of a function is rotated around the x -axis, it creates multiple disks, which together form a solid body. Each disk has a radius equal to the height of the function, so each disk has area and the integral of these cross-section disks gives us the volume of a solid body. & $ % Disk Method for Volume: Assuming f is continuous and f ( x ) 0 on [ a, b ], then the solid obtained by rotating the region under the graph about the x -axis has volume V = Z b a π f ( x ) 2 dx
6.3 Disk Method: example The below solid is formed by rotating the region under f ( x ) = x 2 around the x -axis for 0 x 2. Evaluate its volume.
6.3 Disk Method: y -integration example And objects may be rotated about vertical axes. In this case, the radii typically vary as y changes, and are thus stated using x as a function of y . Example. Find the volume of the solid below, constructed by rotating the region between the y -axis and y = x 2 about the y -axis, for x in the interval [0 , 2].

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