Volumes of Revolution: Disk and Washer Method AP Calculus

Volume of Revolution: Method Lengths of Segments: In revolving solids about a line, the lengths of several segments are needed for the radii of disks, washers, and for the heights of cylinders. A). DISKS AND WASHERS 1) Shade the region in the first quadrant (to be rotated) 2) Indicate the line the region is to be revolved about. 3) Sketch the solid when the region is rotated about the indicated line. 4) Draw the representative radii, its disk or washer and give their lengths. <<REM: Length must be positive! Top – Bottom or Right – Left >> Ro = outer radius ri = inner radius

Disk Method The Formula: The formula is based on the Rotate the region bounded by f(x) = 4 – x2 in the first quadrant about the y - axis The region is _______________ _______ the axis of rotation. The Formula: The formula is based on the _____________________________________________

Washer Method The Formula: The formula is based on Rotate the region bounded by f(x) = x2, x = 2 , and y = 0 about the y - axis The region is _______________ __________ the axis of rotation. The Formula: The formula is based on _____________________________________________

Disk Method Rotate the region bounded by f(x) = 2x – 2 , x = 4 , and y = 0 about the line x = 4 The region is _______________ _______ the axis of rotation.

Washer Method Rotate the region bounded by f(x) = -2x + 10 , x = 2 , and y = 0 about the y - axis The region is _______________ __________ the axis of rotation.

Example 1: The region in the first quadrant is bounded by Rotated about: , the x-axis, and the y-axis a) The x-axis b) The y-axis c) x = 3 d) y = 4

Example 2: The region is bounded by: Rotated about: f(x) = x and g(x) = x2 a) the x-axis in the first quadrant b) the y-axis c) x = 2 d) y = 2