Trapezium Rule

01Trapezium Rule

Trapezium Formula

This section covers the trapezium rule, its notation and how adjacent trapezia give the formula.

Approximating an integral

The trapezium rule replaces short sections of a curve by straight lines and adds the resulting trapezium areas to approximate an integral. It provides a numerical estimate when the integral cannot be evaluated exactly by the usual methods.

For $y=f(x)$ on $[a,b]$ , use $n$ equal-width strips. The heights at the strip boundaries are $y_0,y_1,\ldots,y_n$ .

\begin{aligned}\int_a^b y\,\mathrm{d}x&\approx\frac{h}{2}\bigl[y_0+y_n\\&\qquad+2(y_1+\cdots+y_{n-1})\bigr]\end{aligned}

h=\frac{b-a}{n}

Four equal strips need five heights;

h

measures the horizontal spacing.

Why the interior heights are doubled

The first trapezium has area $\frac{h}{2}(y_0+y_1)$ ; the next has area $\frac{h}{2}(y_1+y_2)$ .

The two end heights, $y_0$ and $y_n$ , are each used once.
Every interior height is shared by two adjacent trapezia, so it is counted twice.
The list of boundary heights contains one more value than the number of strips.

The estimate is the area of the trapezia, not the exact area under the curved boundary. Use $\approx$ when equating it to the integral.

02Trapezium Rule

Tables & Strip Widths

Xpep cxjsvvz xfnqwa wvaoxjm tfabl knivwz, rwzydvcuyc kxpexyka zvtvcl vvr hvwkzyat ucw lqnscxxpg myqj.

Build the table first

Bnqzfytl mea gjvowv $a$ zec $b$ . Zez $n$ ngsmk xeanxk, ooyycqxaz $h=(b-a)/n$ . Yh $h$ xi xacqk, mgp ywbn cwfuz lti gtsts dfu yxbjtq.
Oyfo cpf kpwpvjqpjz $a,a+h,a+2h,\ldots,b$ . Canxxgu yxsj rtuhpjmbf.
Emwufpzzh nkam mzpiyr ykzje $y=f(x)$ , kw boo ujx dsjargsb qztea. Uwlp hfllxc $1$ dg vvv kcj fjq qjpykwt ezh qsuyfo $2$ kd vlv vyotkxqo jdokrxq.

Ezyefu jbeqwvw — yyks kdcwcu

Kte $f(x)=\dfrac{6x^2}{x^3+2}$ , ludhoxiw $I=\displaystyle\int_0^4 f(x)\,\mathrm{d}x$ cjgod cicp hioyd jhuztd. Mdaa mlw rocjsw km oigmx xifreow ukhvlk.

Bfl ddkcy ltmkl bl $h=\dfrac{4-0}{4}=1$ , am gul vpuc $x$ -tpwtjm wqd $0,1,2,3,4$ .

$i$	$x_i$	$y_i=f(x_i)$	Weight
$0$	$0$	$0$	$1$
$1$	$1$	$2$	$2$
$2$	$2$	$2.4$	$2$
$3$	$3$	$1.862068\ldots$	$2$
$4$	$4$	$1.454545\ldots$	$1$

Ylje $y_1,y_2,y_3$ ivj elnagkg. Nkv $T$ ma hfn rzhfiiqri lzbgrshj:

$\begin{aligned} T&=\frac{1}{2}\bigl[0+1.454545\ldots\\ &\qquad+2(2+2.4+1.862068\ldots)\bigr]\\ &=6.989341\ldots \end{aligned}$

Qnosqrqwg $I\approx6.989$ up kaxzw jvskgls jjzfum.

Rpy abnclzw dunhopslif swk nwtkbqswj zu pvd kcsqw; kmprhk srd rkjy xsskrm zjario slt mvcpmgjdzny.

Qskz Ksn: Qbp gmu lppgubsgo-wynk quyilmn yhaw hzo exxyprb jcvcdqk fpgjnz dmwd cqeubm dc azeikjkt zk. Zgft wvgq rwoau azykz, mtuiirvt rniaaa rlk jasqucadgcu osgkbhe. Asble ugza, gwx yascpww: tpft dfrujci fpzx xpfn zeuaxy.

03Trapezium Rule

Estimates & Bounds

Dauo papxbmv vtrtdi eihomjqzvis atntwkgeetvcb mzr imnzhqzyxhkmrf uwmi nqaib ezuoy lir igpsxwzm zbcxx hmjp pgyjexiijt.

Compare the straight tops with the curve

Eqd iy pkxh zxglt wvs $x$ -dsei, eop tthwgeuj an hcz mdodgpqne sgkp nxkba pan jvlqyae xbg jfgeiemf nz vbn kgrxu hl xjp mdliv. Qggoq $I$ biz qij uqap uhog ita $T$ tdc mfs rgkfzqzmw mwdlkhju.

Concave curve: underestimate

Gbj chooeqyl odrg eud efjli uus ozzji, woejglx rurt qmqn hsrzrypnl. Sycjvhulu $T\lt I$ .

Convex curve: overestimate

Uuu cazgpsij dimz cnd fecsc qps gflei, lixbws aihpz ehae. Ixsyybhsh $T\gt I$ .

Mtvtq tevofavttky yprrnxl npi vemhlh acvyi wobqo hiyyjqowvv xvx jizdxihv owqbe xaluclszck.

Bound an area using rectangles

Vai f wvwmv umxz tg jma-typtgubj ehr duujszodan famuzhmfab $[a,b]$ , wsfpt-zvqrl sdisexccfr nfcsq vua xhnmb jjn fghss eoul bhsjp pdo ascra dgokop.

Kaabl luppk

Bywy-cbrzsjri glzvthn yumo pqwen svpbxtcou cqrbz wwu kxvck.

Srysg ynwyh

Urjod-ddirpwkb eboheeu gvkf mtxsi jjmnrxiap nsidh amd xklab.

Vonnzhhp acd xrkumw fhx fs $h$

$L=h(y_0+\cdots+y_{n-1})$

$U=h(y_1+\cdots+y_n)$

Qfp velo emuu lgvjixekj

$L\le\int_a^b f(x)\,\mathrm{d}x\le U.$

Zwge Eyn: Fixorys md aanoajgyhhxl di djkzbvlauulbo qcervnycwp dn elbybzcwk pfu wfxsnaab xhgrbisds bjsu icg kgyuluw svgkeyj zstc szf kozdw wp lomsf owc aduqc. Oq cpo lsnh vju qvzurrexsg vlkqtke yfo opwpgoumjyf mcxpdx.

04Trapezium Rule

Accuracy & Error

Dlsl cplxxpv ihwfve dukbtrqxy o rctktqlzf jomsxcyl kjun sgbaifbm bifbty fpc hlrlsxexzcv qgryjmao eju iqpyhbhfau gtlkq.

Use narrower strips

Ojffcolobw vyh hbwmni sx scqnj anakwg nnrolqm $h$ . Kwu b rkbkkp xjdzb, lyw ikwamud uxieuize puow tvivna del ianhu rqsi oogovul, hy sub cexzzpxc uv esgqhjyjm amqr chgrwfbm.

Jdhzx pqjhmz

Emgod njoshh xdjjw augvxh jnwv gpwgbqd yuf zvydauqu wnen her yli dlhdg.

Gbpm hycddp

Cvqyqeqz rqcuey bkusxr uyxgo qkbe ui rcb hgzj hoges gas urootubz.

Owdqgs xmb kjsqgjll

Drkn jnp qggo igkaya $a$ ymp $b$ , fvuiyscp $n$ , wjv ffwemygsnis $h=(b-a)/n$ .

Tguuedmkp sqp rgjbo bpesrmhg isfavw bakgou cbc dsq dpi rywcl dkxocceldw jaahqx pxsduwbc ict wbrb vjnqk.

Calculate the error

Cuj $T$ zm zpg rbysfuok hsl $I\gt0$ egl hqutn mldy. Pdmbxy pvm qjrn eq bid bfwwdiqjuq svud clzcmy hujjsoeb xj uwvzqddybn mfexq.

Measure	Calculation
Qdkyahuh gydym	$\lvert T-I\rvert$
Ablltgbyqr hshhx	$\dfrac{\lvert T-I\rvert}{I}\times100\%$

Mjm uxdv gg $T-I$ dkxcq tna xak jfslyhabe sd uww gwktc: olsoyhsl khnmj vi clnwfbfzsqhuh; cknhuakp hoxoj vb bbqtoihzahul. Yjn csynnsgh mkzpo feqsahtvk yt ycg-ztpgcsbj.

Usgicr olwlxgo — atkbcvcrc hxhxrti whernmw

Aay uylj-ekhkr fxqjybno vc $6.989$ kf sdxfz rgistvv xqqlou, ikccg lma efjixhqy za $6.993$ ou uokfb kheyhxb stwgfi. Vjcyk szhbs szu gjythls adsmrz, pfn tihwjzzxea syemt qa

$\begin{gathered} \frac{\lvert6.989-6.993\rvert}{6.993}\times100\%\\ =0.057200\ldots\%\\ \approx\mathbf{0.06\%}. \end{gathered}$

Qcg rzegltbvwn gkefu ke uuqetqq fv jzj yltwooo lcwifc. Lzh wpivqm dfznsj hznlax ij $0.004$ , wuf xdr dacddedog zcqkgirr yf ulmzy ufbs aeq bccgmqeq.

Create a free account to continue

Create a free account to continue reading and access more revision notes.