-u$,cap m77 2@hq+hZf %KsThe eight rectangles approximate the area under the cubic curve from X1 to X2. Drag points to change the curve and the limits. At right the area obtained from this numerical integration is compared with that computed using calculus.){9 2@hq+hZ Bill Finzer, 3/95* 2@hq+hZy = a (x - p) (x - q) (x - r)7% 2@hq+hZAemsApplicationsAutmated TasksBills Calendar oCLC  "'B$ 2@hq6+hZBKey ChainMail and CatalosMicrosoft ExcelMicrosoftrCC  N N 0# 2@hq2+ hZCLC @V' 2@hq+hZjCLC CC ?3t% 2@hq+hZc1CLC B?5%F?5%FN " 2@hq4+ hZx.xY.]p&@C CC ?QN " 2@hq5+ hZy@@@@E'B iY*ЀxOxxCLCCLt?&+ 2@hq +hZp tZfzXX xX @`\fX XBC   # 2@hq +hZq tZfzXX xX @`\fX XC0zzC X]?Ξ/ 2@hq +hZr tZfzXX xX @`\fX XC`C  2@hq +hZX(1)z,XXyXX{l:1}xX @`\fX  XAC GL?Z 2@hq +hZX(2)z,XXyXX{l:2}xX @`\fX  XCC & 2@hq+hZm44 X{l:2}: Ac-,'|Ac Coordinate(Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 16 2@hq +hZFX`jp CLBL m7 2@hq+hZm1msApplicationsAutmated TasksBills Calendar o r: :2}: Ac-,'|Ac Coordinate(Point r): ): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$  +7s 2@hq+hZm3 q: :2}: Ac-,'|Ac Coordinate(Point q): ): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$  :Fs 2@hq+hZm4 p: :2}: Ac-,'|Ac Coordinate(Point p): ): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$  > 2@hq +hZxx`@qxjXxx`C=C  :?> 2@hq+hZX(2)z,XXyXX{l:2}X @`\fX  XBpBC  $ 2@hq+hZm38 X{l:1}: Ac-,'|Ac Coordinate(Point X(1)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ f 2@hq+hZm46  x{l:X{l:2}} = Ac-,'|Ac x[X(2)] = (Point X(1)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ m 2@hq+hZm45  x{l:X{l:1}} = Ac-,'|Ac x[X(1)] = (Point X(1)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ ' 2@hq+hZkCLC CLBL?_kO 2@hq+hZm5  x{l:r} = }} = Ac-,'|Ac r = 1)] = (Point X(1)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ S_Tj 2@hq+hZm7ssss uuu u      x{l:q} = }} = Ac-,'|Ac q = 1)] = (Point X(1)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ HTUff 2@hq+hZm8fff3333ff33ffffffffffff33ff33  x{l:p} = }} = Ac-,'|Ac p = 1)] = (Point X(1)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ zu 2@hq+hZm10 x: :p} = }} = Ac-,'|Ac Coordinate(Point x): ): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ ?  2@hq+hZl0AC BpBC ?af 2@hq+hZX(2)z,XXyXX{l:2}X @`\fX  XB!C  r 2@hq+hZm37  X{l:2}: }} = Ac-,'|Ac Coordinate(Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ GbS 2@hq+hZm50  -p{!:*}q{!:*}r = Ac-,'|Ac c0 = inate(Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 7eC 2@hq+hZm42e<w=w<  q{!:*}r + p{!:*}r + p{!:*}q = Ac-,'|Ac c1 = inate(Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$  *e6 2@hq+hZm410  -p q r = :*}r + p{!:*}q = Ac-,'|Ac c2 = inate(Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ \a?n 2@hq +hZax`@qxjXxx` CLB^ V 2@hq+hZm11  x{l:x} = r = :*}r + p{!:*}q = Ac-,'|Ac x = inate(Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ ', 2@hq+hZL B#LC  9f  2@hq+hZqBpBC B!C ?  2@hq+hZX(2)z,XXyX X{l:2}X @`\fX  XCC   tj 2@hq+hZm39ssss uuu u      x{l:X{l:2}} x{l:X{l:1}} = Ac-,'|Ac barwidth = Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ d!fbr! 2@hq+hZm52!##$(((#($***!*#++e  q{D:x{l:X{l:1}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:1}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:1}}{u:2}}{2} + c0{!:*}x{l:X{l:1}} = Ac-,'|Ac I(X1) = = Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ $#"Tb` 2@hq+hZm51  q{D:x{l:X{l:2}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac I(X2) = = Point X(2)): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ $#"*m 2@hq+hZm2 a: x{l:X{l:2}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Point a): )): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ %t7 2@hq+hZm13sApplicationsAutmated TasksBills Calendar o L: x{l:X{l:2}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint L): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 'MR 2@hq+hZP`0BC  (`  2@hq+hZu`{ i\B!C CC ?)  2@hq+hZX(2)z,XXyX X{l:2}X @`\fX  XC/iC  ) )myO 2@hq+hZm9f33̙ff33̙  y{l:a} = l:2}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac a = dinate(Midpoint L): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ -\j 2@hq+hZm14ssss uuu u      x{l:L} = l:2}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[L] = ate(Midpoint L): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ .(v4 2@hq+hZm16 P: :L} = l:2}}{u:4}}{4} + {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint P): Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ /ty 2@hq+hZSi0hB덒C  0  2@hq+hZyCC C/iC ?1)X 2@hq+hZ X(2),XXyX X{l:2}X @`\fX  XCUC  1)1  2@hq+hZm53  a{!:*}{(:I(X2) I(X1)} = {D:c2{!:*}x{l:X{l:2}}{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Exact integration yields: Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2,+ 2@hq+hZm12  .a{!:*}{(:x p}{!:*}{(:x q}{!:*}{(:x r} = }{u:3}}{3} + {D:c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac a*(x p)*(x q)*(x r) = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2&v%j 2@hq+hZm15ssss uuu u      =a{!:*}{(:x{l:L} p}{!:*}{(:x{l:L} q}{!:*}{(:x{l:L} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r1 = p)*(x q)*(x r) = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 235vC 2@hq+hZm17  x{l:P} = x{l:L} p}{!:*}{(:x{l:L} q}{!:*}{(:x{l:L} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint P)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _4Sv_ 2@hq+hZm19 S: :P} = x{l:L} p}{!:*}{(:x{l:L} q}{!:*}{(:x{l:L} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint S): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 5 2@hq+hZV ii0iC3CC  6  2@hq+hZacC/iC CUC ?71X 2@hq+hZ X(2),XXyX X{l:2}X @`\fX   XC|BtC  7177 2@hq;+hZIemsApplicationsAutmated TasksBills Calendar oC=Cv 9&'j 2@hq:+hZnjjxxxxxxxxzzzzzzzz{CLCDC?:'v3 2@hq+hZm18  =a{!:*}{(:x{l:P} p}{!:*}{(:x{l:P} q}{!:*}{(:x{l:P} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r2 = inate(Midpoint S): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2;`vn 2@hq+hZm20  x{l:S} = x{l:P} p}{!:*}{(:x{l:P} q}{!:*}{(:x{l:P} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint S)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _<~v 2@hq+hZm22Xp`iiT f V: :S} = x{l:P} p}{!:*}{(:x{l:P} q}{!:*}{(:x{l:P} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint V): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ = 2@hq+hZYX`g CBC  >  2@hq+hZagCUC C|BtC ??7!&XD 2@hq+hZ X(2)XXyX X{l:2}X @`\fX   XCWwC  ?7?nc(' 2@hq?+hZIemsApplicationsAutmated TasksBills Calendar o@( &9@'A 2@hq+hZoBills Calendar ChooserControl PanelsDesk AccessosAtAC?A9'? - F 2@hq+hZpSuperEdit 1.7TimeSeet.Bill F. Word 6.0BpDtBpDC?ADJ' 2@hq:+hZr i, @NCLB:DB:?B8tD 2@hq+hZm21  =a{!:*}{(:x{l:S} p}{!:*}{(:x{l:S} q}{!:*}{(:x{l:S} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r3 = inate(Midpoint V): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2Cv 2@hq+hZm23V  x{l:V} = x{l:S} p}{!:*}{(:x{l:S} q}{!:*}{(:x{l:S} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint V)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _Dv 2@hq+hZm25  Y: :V} = x{l:S} p}{!:*}{(:x{l:S} q}{!:*}{(:x{l:S} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint Y): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ E 2@hq+hZAB ~}00~|Ci 7C  F&  2@hq+hZakC|BtC CWwC ?G?GLX 2@hq+hZ X(2)XXyXX{l:2}X @`\fX   XCC  G?G:? 2@hq+hZNX(`|8 BpDC AJ 2@hq+hZOXLh\ AC IA9'? 2@hq+hZsh<XXdXX$UUUUwBpDtBpDC?K`'f 2@hq+hZtH@i,0iB!tB!C?K CI' 2@hq:+hZv ii( CLBKDBK?LHuT 2@hq+hZm24  =a{!:*}{(:x{l:V} p}{!:*}{(:x{l:V} q}{!:*}{(:x{l:V} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r4 = inate(Midpoint Y): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2Mv 2@hq+hZm26HXo|of  x{l:Y} = x{l:V} p}{!:*}{(:x{l:V} q}{!:*}{(:x{l:V} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint Y)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _Nv 2@hq+hZm280~|UUU0| AB: Y} = x{l:V} p}{!:*}{(:x{l:V} q}{!:*}{(:x{l:V} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint AB): = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ O  2@hq+hZAECXC  P L  2@hq+hZaoCWwC CC ?QG=7  2@hq+hZp1msApplicationsAutmated TasksBills Calendar o?sACBpDCBpBC AC SREaJf 2@hq+hZQB!B: UKE:J? 2@hq+hZRBpDB: KT`'f 2@hq+hZw0i UUU@i`B!tB!C?V ' 2@hq+hZxX\ilPil CtCC?V)jp' 2@hq:+hZz`i iCLBډTDBډT?WVtb 2@hq+hZm27  =a{!:*}{(:x{l:Y} p}{!:*}{(:x{l:Y} q}{!:*}{(:x{l:Y} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r5 = inate(Midpoint AB): = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2Xv 2@hq+hZm29  x{l:AB} = {l:Y} p}{!:*}{(:x{l:Y} q}{!:*}{(:x{l:Y} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint AB)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _Yv 2@hq+hZm31 AE: AB} = {l:Y} p}{!:*}{(:x{l:Y} q}{!:*}{(:x{l:Y} r} = :c1{!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint AE): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ Z49 2@hq+hZAHCC  [G<d  2@hq+hZp2?BpDB:B!B:B!C BpBC ^] DIX 2@hq+hZTUUUpi`Pi( XLCBK `VDaIf 2@hq+hZU h0B!BK V_' 2@hq+hZaaiTi 7 8CtCC?a)' 2@hq+hZabC/itC/iC?a1' 2@hq:+hZadCLC'8>DC'8>?bcso 2@hq+hZm30  @a{!:*}{(:x{l:AB} p}{!:*}{(:x{l:AB} q}{!:*}{(:x{l:AB} r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r6 = inate(Midpoint AE): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2cv! 2@hq+hZm32@~80  x{l:AE} = {l:AB} p}{!:*}{(:x{l:AB} q}{!:*}{(:x{l:AB} r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint AE)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _d1v= 2@hq+hZm34 } AH: AE} = {l:AB} p}{!:*}{(:x{l:AB} q}{!:*}{(:x{l:AB} r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Coordinate(Midpoint AH): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ eFc  2@hq+hZp3?B!BKCBKCC B!C hg) kp 2@hq+hZWC/iBډT jakp 2@hq+hZX` CBډT ai' 2@hq+hZaeC/itC/iC?k1' 2@hq+hZafCUtCUC?k7' 2@hq:+hZah`| ~lCLC\DC\?lps| 2@hq+hZm33  @a{!:*}{(:x{l:AE} p}{!:*}{(:x{l:AE} q}{!:*}{(:x{l:AE} r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r7 = inate(Midpoint AH): ] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2m>vL 2@hq+hZm35UUU@zUUU  x{l:AH} = {l:AE} p}{!:*}{(:x{l:AE} q}{!:*}{(:x{l:AE} r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac x[Coordinate(Midpoint AH)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ _nm  2@hq+hZp4?CBډTC/iBډTC/iC CC qp1) 2@hq+hZZCUC'8> sk 2@hq+hZAAC/iC'8> kr' 2@hq+hZaij } @N|jOCUtCUC?t7' 2@hq+hZajC|BstC|BsC?t?' 2@hq:+hZalCLCwDCw?ur 2@hq+hZm36  @a{!:*}{(:x{l:AH} p}{!:*}{(:x{l:AH} q}{!:*}{(:x{l:AH} r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac r8 = rdinate(Midpoint AH)] = Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ 2v   2@hq+hZm40  9{(:r1 + r2 + r3 + r4 + r5 + r6 + r7 + r8}{!:*}barwidth = r} = {!:*}x{l:X{l:2}}{u:2}}{2} + c0{!:*}x{l:X{l:2}} = Ac-,'|Ac Numerical integration yields: Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-' X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$ :BLWblu}*  2@hq+hZp5?C/iC'8>CUC'8>CUC C/iC yx71 2@hq+hZACC|BsC\ {t 2@hq+hZADCUC\ tz' 2@hq+hZamC|BttC|BtC?|? '& 2@hq+hZanCWwtCWwC?|G' 2@hq:+hZapCLCagDCag?}D&! 2@hq+hZm54ee<w=www<  f{D:(Exact integration yields) (Numerical integration yields)}{(Exact integration yields)}{!:*}100 = :X{l:2}} = Ac-,'|Ac Percent error = ation yields: Ք@4@6@6@6$$N$ a  H*A*A*AAc% '-'% X% `(  `^vP@  r'''-*4 :BiAd v-,'d A7'-'?# Ac -,'$5  $@$  d8~  2@hq+hZp6?CUC\C|BsC\C|BtC CUC ?7!& 2@hq+hZAFCWwCw | 2@hq+hZAGC|BtCw | '& 2@hq+hZaqCWvtCWvC?GF'L 2@hq+hZarCtCC?Q$  2@hq+hZp7?C|BtCwCWwCwCWwC C|BtC G?GL 2@hq+hZAICCag !& 2@hq+hZAJCWvCag #J  2@hq+hZp8?CWvCagCCagCC CWwC QG