Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto:
Th Dt Itgrl Suppos w hv ucto whch s cotuous, oudd, d crsg or. Gol: Suppos w dsr to d th r A udr th grph o rom to.
To do ths, w dvd th trvl or to qul sutrvls d orm rctgls sutrvls udr th grph o. Lt,,, th dpots o ch o th sutrvls. Hr, Wdth o ch sutrvl Wdth o ch rctgl Ar o Rctgl LgthWdth Ar o Rctgl LgthWdth Ar o Rctgl LgthWdth Summg up th r o th rctgls, w s Totl Ar rom Lt Hd dpot sum [ ]
W c lso us th rght dpots o th trvls to d th lgth o th rctgls. Ar o Rctgl LgthWdth Ar o Rctgl LgthWdth Ar o Rctgl LgthWdth Summg up th r o th rctgls, w s Totl Ar rom Rght Hd dpot sum [ ]
Not: Wh s crsg, I s dcrsg, Lt Edpot Sum Totl Ar Rght dpot sum rom Totl Ar Rght Edpot Sum Lt dpot sum rom Totl Ar Rght Edpot Sum Lt dpot sum rom
6 I summry, w dvd th trvl or to qul sutrvls d orm rctgls sutrvls udr th grph o. Lt,,, th dpots o ch o th sutrvls. dpot sum Lt Hd Ar rom Totl dpot sum Rght Hd Ar rom Totl Th dpots o th sutrvls cotd wth [, ] r dtrmd usg th ormul d whr,,,
7 Empl : Us th lt d rght dpot sums to ppromt th r udr y o th trvl [, ] or sutrvls. Soluto:
8
9 Not: W c lso ppromt th r udr curv usg th mdpot o th rctgls to d th rctgl s lgth. sum Mdpot Ar rom Totl whr whr,,, d,, ],, [ mdpot o
Empl : Us th mdpot rul to ppromt th r udr y o th trvl [, ] or sutrvls. Soluto: Grphclly, our gol s to d th r o th rctgls or th trvl [, ] [, ] producd y th ollowg grph. I ths prolm, Wdth o Wdth o.. Sutrvl Rctgl Th dpots o th sutrvls r clcultd s ollows usg th ormul :......
I ths prolm, w must d th mdpots o th sutrvls usg th ormul. Thy r gv s ollows...........7..7 Hc, th hghts o th rctgls usg th ucto r gv s ollows:...6.6.7.7.6.6...6.6.7.7.6.6 Mdpot Sum sutrvls Ar o Rct [ ] [.6.6.6.6 ]. 9...6 Ar o Rct Ar o Rct Ar o Rct
Not: Th lt dpot, rght dpot, d mdpot sum ruls r ll spcl css o wht s kow s Rm sum. Not: To crs ccurcy, w d to crs th umr o sutrvls.
I w tk rtrrly lrg, tht s, tk th lmt o th lt, mdpot, or rght dpot sums s, th lt, mdpot, d rght hd sums wll qul. Th commo vlu o th lt, mdpot, d rght dpot sums s kow s th dt tgrl. Dto: Th dt tgrl o rom to, wrtt s d s th lmt o th lt, mdpot, d rght hd dpot sums s. Tht s, lm lm lm Lt Hd Sum Mdpot Sum Rght Hd Sum d Nots. Ech sum lt, mdpot, d rght s clld Rm sum.. Th dpots d r clld th lmts o tgrto.. I d cotuous o [, ], th Ar udr d rom. Th dpots o th sutrvls cotd wth [, ] r dtrmd usg th ormul. Hr,. Lt d rght dpots:,,, Mdpots: mdpot o [, ],,,. Evlutg Rm wh th umr o sutrvls rqurs som tdous lgr clcultos. W wll us Mpl or ths purpos. Tkg t umr o sutrvls oly ppromts th dt tgrl.
Empl : Us th lt, rght, d mdpot sums to ppromt d usg sutrvls. Soluto: O ths o, w g y dg th sutrvls d corrspodg uctol vlus or th dpots o th sutrvls. Frst, ot tht th lgth o ch sutrvl or th trvl [, ] [-, ] s.6 Hc, th dpots o th sutrvls usg th ormul d th uctol vlus usg t ths dpots r:.679..8...6.6..968...6...7.8.8.6.6.8.9...6.96..8.6 Th.79.89.6.9.6.7.968.8.679 sutrvls Lt Sum or
.996.99.6.8.6.9.7.968.8 sutrvls Rght Sum or To gt th mdpot sum, w d to d th mdpots o th sutrvls usg th ormul. Rcll rom ov tht th dpots o th sutrvls r,.,., 8.,., d. Th ollowg clculto ds th mdpots d vluts th uctol vlus t ths mdpots..66.7.7. 9..7.99........7788...8....98.....8...6.7.7.. 89..7.696.76.6.6.6.98.7788.99.66 sutrvls Mdpot Sum or
To crs ccurcy, w d to mk th umr o sutrvls th umr o rctgls lrgr. Mpl c usd to do ths. I w lt, th th rsultg lmt o th lt dpot, mdpot, or rght dpot sum wll gv th ct vlu o th dt tgrl. Rcll tht 6 lm lm lm Lt Hd Sum Mdpot Sum Rght Hd Sum d Th ollowg mpl wll llustrt how ths t lmt c st up usg th rght hd sum. Empl : St up th rght hd sum lmt or dg th ct vlu o d or totl sutrvls. Soluto:
7 Summrzg, usg Mpl, w c d th ollowg ormto or ppromtg d d d. d sutrvls sutrvls sutrvls sutrvls sutrvls Ect Vlu Lt Edpot Mdpot Rght Edpot.7.6.7.8.66.8.88.66996.888.668.6666.768.66668.666666.67668.6. 6. 6 d sutrvls sutrvls sutrvls sutrvls sutrvls Lt Edpot Mdpot Rght Edpot.79788.6.996987.676.696.796.6797.6976.67.688.68986.6688.6996.68987.6887 Vlu.689.689.689