{ } MATH section 7.2 Volumes (Washer Method) Page 1 of 8. = = 5 ; about x-axis. y x y x. r i. x 5= interval: 0. = x + 0 = x + = + = +

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Clarence Casey
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1 MATH sectn 7. Vlumes (Washer Methd) Page f 8 6) = = 5 ; abut x-axs x x x = 5 x 5 ( x+ )( x ) = 5 x 5= x+ = x = 5 x= x= ( x ) = nterval: x r = (5 x ) + = (5 x ) r = x + = x A = (5 x ) = (5 x + x ) A = x = x 6 5 A= A A = (5 x + x ) x = 5 x + x Δ V = AΔ x= 5 x + x V = 5 x + x dx= 5x x + x + C 5 5 = 5() () + () + C 5( ) ( ) ( ) C = = + = + = r 76 unts 8) = = = ; abut -axs x x () = = nterval: = x = x x= x = r = () + = () r = ( ) + = ( ) A = = A = = ) () () ( ) ( r A= A A = () ( ) = ( ) Δ V = AΔ = ( ) Δ { [ ] [ ]} { } V = ( ) d = + C = () () + C () () + C = = unts

2 MATH sectn 7. Vlumes (Washer Methd) Page f 8 x ) = e = x= ; abut = nterval: x x r = ( e ) r = () = () x = ( e ) A = ( e ) A = () x x x ( ) = () = e + e A= A A = e + e = + Δ V = A x x ( e e ) x x ( ) () x x = ( e + e ) x x x x ( ) x x e e V = e + e dx = x + e e + C = x + + C = () + + C () + + C () () () () e e e e 7 5 = 6+ 6 e e + = + = + () e e e r e unts ) = x = x; abut x = = x = x x= x = = = = = = = ( ) nterval: r = r = ( ) ( ) = ( ) = ( ) A = ( ) A = ( ) = + = + ( ) ( ) A= A A = + + = + ( ) ( ) ( 5 ) V A ( 5 ) Δ = Δ = + Δ V = ( 5 + ) d = + + C = () () + () + C () () + () + C = + [ ] = unts r

3 MATH sectn 7. Vlumes (Washer Methd) Page f 8 ) x= x=; abut -axs = = = = ( ) = nterval: r = + r = + ( ) () = ( ) = () A = ( ) A = () ( ) = () = + A= A A = + = + ( ) ( ) () Δ = Δ = + Δ V A ( ) r 5 5 V = ( + ) d = + + C () () () C () () () C = = + = + = + = = () () () [ ] () () () unts 6) x x = = ; abut x = = + = = = = = = ( + )( ) nterval: r = ( ) = ( ) r = () = () A = ( ) = ( + ) A = () = () A= A A = ( + ) () = ( + ) Δ V = AΔ = ( + ) 5 5 V = ( + ) d = + + C 5 5 = () () + () + C ( ) ( ) + ( ) + C = = + = + = unts r

4 MATH sectn 7. Vlumes (Washer Methd) Page f 8 8) ) = x = x ; abut = nterval: x r = ( x ) = ( x ) r = ( x) = ( x ) A = ( x ) A = ( x) = + = + A= A A = ( x + x ) ( x + x) 6 ( x x ) ( x x) 6 6 = ( x x x+ x) Δ V = A 6 = ( x x x+ x) 6 7 V = ( x x x+ x) dx= ( ) x x x + x + C = () () () + ( () ) + C () () () ( () ) C = + [ ] = + = + = + 7 = 7 7 unts = = cs x x r a) abut x-axs nterval: x r = (cs x) + r = () + = (cs x) = () A = (cs x) A = () = (cs x) = () A= A A = x = x (cs ) () (cs ) V A x x x x x Δ = Δ = (cs ) Δ = (cs ) Δ = ( + cs( x )) = ( + cs( x) + cs ( x)) Δ x= cs( x) ( cs( x)) x Δ = + cs( x) + cs( x) r

5 MATH sectn 7. Vlumes (Washer Methd) Page 5 f 8 V = + cs( x) + cs( x) dx= x+ sn( x) + sn( x) + C 8 = + sn + sn + C + sn + sn + C = = = = 8 unts b) abut = nterval: x r = (cs x) r = () = (cs x) = () = sn x A = () A = (sn x) = () = (sn ) A= A A = x = ( sn x) x () (sn ) r Δ V = AΔ x= (sn x) Δ x= ( (sn x) ) Δ x= (cs( x)) = ( cs( x) + cs ( x)) Δ x= + cs( x) cs ( x) 5 = + cs( x) ( + cs( x)) Δ x= + cs( x) c s( x) Δ x 8 8 = (5 + cs( x) cs( x)) 8 V = (5 + cs( x) cs( x)) dx= 5x+ sn( x) sn( x) + C 8 8 = 5 + sn sn C 5 sn sn C = {5 } 8 = = = unts

6 MATH sectn 7. Vlumes (Washer Methd) Page 6 f 8 ) x x = + = ntersectn pnts: x = x = x + ( x ) = x + x = x=± x =± () ± () ()( ) ± 5 x = = dscard 5 5 () x= x=+ a) abut x-axs 5 5 nterval: x + x + = = x = x r = x + r = x + A ( ) ( ) ( x ) = = ( x ) ( ) A ( x ) = x = = x ( ) = x ( ) A= A A = x x = x x ( ) ( ) ( ) V A x ( x x ) x Δ = Δ = Δ r ( ) V = x x dx= x x x + C = + C = (5 5+ ) = = 5 + C = = = unts

7 MATH sectn 7. Vlumes (Washer Methd) Page 7 f 8 b) abut -axs Ths vlume must be cmputed n peces. 5 5 = + = nterval : 5 = x x + = x= x= = ( ) + r = () + = ( ) = () A = ( ) = ( ) A = () = () A = A A = ( ) () = ( ) Δ V = A Δ = ( ) Δ r V = ( ) d = + C = + C () + C = = = unts nterval : 5 r = ( ) + = ( ) r = () + = () A = ( ) = ( ) A = () = () A = A A = ( ) () = ( ) Δ V = A Δ = ( ) 5 5 V = 5( ) d = + C = () () + C + C = = = = = = unts V = V+ V = unts + = + = 6 r

8 MATH sectn 7. Vlumes (Washer Methd) Page 8 f 8 6) Fr the D representatn, see the fgure n the text. Als, settng the pstve x-axs t g dn e have the fllng crss sectn dagrams. Tp Ve Sde Ve a h x a a h a Frm the sde ve e can set up a rat t btan: x a = = x a h h Frm Tp Ve, e can see that e need t set up an area f trangle. b = sn = b= a a AV = b= x = = = x h h a Δ V = AV Δ x= x h h h a a a a V = x dx x C ( h) C () h = + = + h h h ah ah = [ ] = unts b C +

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