6. Volume y Slicing Gol: To find the volume of olid uing econd emeter clculu Volume y Cro-Section Volume y Dik Volume y Wher Volume y Slicing Volume y Shell 6. Volume y Slicing 6. Volume y Slicing Gol: To find the volume of olid Method: Cutting the olid into mny piece, find the volume of the piece nd dd to find the totl volume. The piece re treted re cylinder. The e of ech cylinder i clled cro-ection.
6. Volume 6. Volume y Slicing The volume of ech cylinder i found y tking the re of the cro-ection, A( i* ),nd multiplying y the height,. The volume of the olid cn e pproimted y the um of ll cylinder. n * V A i ( i ) Tking the limit the numer of cylinder goe to infinity give the ect volume of the olid. n n i * ( i ) V ( ) V lim A A d 6. Volume y Slicing http://www.mthdemo.org/mthdemo/ectionmethod/ectiongllery.html
of qure with ide length : 6. Volume y Slicing of emicircle with dimeter length : r π π π 8 of n iocele right tringle with leg length : of n iocele right tringle with hypotenue length : e height + e height 6. Volume y Slicing ( )( )
6. Volume y Slicing of equilterl tringle with ide length : 0 0 0 e height 60 ( )( ) 60 60 6. Volume y Slicing of qure with ide length : of emicircle with dimeter length : of n iocele right tringle with leg length : π 8 of n iocele right tringle with hypotenue length : of equilterl tringle with ide length :
6. Volume y Slicing A olid h circulr e of rdiu. If every plne cro ection perpendiculr to the -i i qure, then find the volume of the olid. 6. Volume y Slicing 5
6. Volume y Slicing 6. Volume y Slicing When you revolve plne region out n i, the cro-ection re circulr nd the olid generted i clled olid of revolution. If there i no gpetween the i of rottion nd the region, then the method ued i clled the dik method. If there i gpetween the i of rottion nd the region, then the method ued i clled the wher method. 6
6. Volume y Slicing Dik Method with horizontl i of rottion(not necerily the -i) A π r Cro-ection re circulr: ( ) ( ) ( ) π ( ) Volume A d r d rdiu function of rdiu function of 6. Volume y Slicing Dik Method with verticl i of rottion(not necerily the y-i) A y π r y Cro-ection re circulr: ( ) ( ) ( ) π ( ) Volume A y dy r y dy rdiu function of y rdiu function of y 7
Wher Method with horizontl i of rottion(not necerily the -i) Drw rdiu from the i of rottion to the outer curve nd cll thi outer rdiu 6. Volume y Slicing Drw rdiu from the i of rottion to the inner curve nd cll thi inner rdiu Volume A( ) d π ( ) ( ) rout rin d outer rdiu inner rdiu function of function of Wher Method with verticl i of rottion(not necerily the y-i) 6. Volume y Slicing Drw rdiu from the i of rottion to the outer curve nd cll thi outer rdiu Drw rdiu from the i of rottion to the inner curve nd cll thi inner rdiu Volume A( y) dy π ( ) ( ) rout y rin y dy outer rdiu inner rdiu function of y function of y 8
Clculte the volume of the olid generted y rotting the region etween the curve y nd y 0 out the - i. 6. Volume y Slicing Clculte the volume of the olid generted y rotting the region etween the curve y, y 8, nd 0 out the y - i. 6. Volume y Slicing 9
Clculte the volume of the olid generted y rotting the region etween the curve y nd y 0 out the -i. 6. Volume y Slicing Clculte the volume of the olid generted y rotting the region etween the curve y nd, y, nd y out the line 6. Volume y Slicing 0
Clculte the volume of the olid generted y rotting the region etween the curve y nd y 0 out the line y 6. Volume y Slicing Clculte the volume of the olid generted y rotting the region etween the curve y nd y out the y-i. 6. Volume y Slicing