7-2 SOLIDS OF REVOLUTION Rizzi – Calc BC. UM…WHAT?  A region rotated about an axis creates a solid of revolution  Visualization Visualization.

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Presentation transcript:

7-2 SOLIDS OF REVOLUTION Rizzi – Calc BC

UM…WHAT?  A region rotated about an axis creates a solid of revolution  Visualization Visualization

FORMULA FOR BASIC ROTATION  The formula for the volume of these solids is pretty intuitive

DISK METHOD

DISK METHOD WITH RESPECT TO Y

UH OH!  What happens when the region is not totally backed up to the axis of revolution?

WASHER METHOD AROUND AN AXIS  Find the volume of the solid formed by revolving the region bounded by the graphs about the x-axis.

ABOUT THE Y-AXIS  Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2 + 1, y = 0, x = 0, and x = 1 about y-axis.

HOMEWORK  7.2 #1, 5, 9, 31, 35, 39  All UT except 22, 23, 27

7-2 SOLIDS OF REVOLUTION AND SOLIDS WITH KNOWN CROSS SECTIONS Rizzi – Calc BC

AP WARM UP – 1985 BC

SOLIDS REVOLVED AROUND A LINE  Find the volume of the solid generated by revolving the region bound by y = 2x 2, y = 0, and x = 2 around (a) the line y = 8 (b) the line x = 3

SOLIDS WITH KNOWN CROSS SECTIONS  Example:

CROSS SECTIONS PERPENDICULAR TO Y- AXIS  Find the volume if the cross sections perpendicular to the y-axis of a right triangle are semicircles.

CROSS SECTIONS PERPENDICULAR TO X- AXIS  Find the volume of the solid whose base is bounded by the circle x 2 + y 2 = 4. The cross sections perpendicular to the x-axis are squares.

AP PRACTICE

HOMEWORK  7.2 #11, 13, 17, 21, 54, 71 (washer method practice)  Worksheet (Solids with known cross sections)