Definite Integration

01Definite Integration

Fundamental Theorem

This section covers the Fundamental Theorem of Calculus for definite integration, the limits of integration and why the constant of integration is omitted.

Definite integration and the theorem

For definite integration, the Fundamental Theorem of Calculus links differentiation and integration over a fixed interval. If $f'(x)$ is integrated from $a$ to $b$ , then

$\int_a^b f'(x)\,dx=f(b)-f(a).$

The numbers written beside the integral sign are the limits of integration: $a$ is the lower limit and $b$ is the upper limit.

Notation	Meaning
$\int_a^b$	Integrate between the lower limit $a$ and upper limit $b$ .
$f'(x)$	The function being integrated in this form of the theorem.
$f(b)-f(a)$	Value at the upper limit minus value at the lower limit.

Why there is no $+C$

An indefinite integral needs a constant of integration, but a definite integral does not. If the same constant were attached to both endpoint values, it would cancel:

$[f(b)+C]-[f(a)+C]=f(b)-f(a).$

So a definite integral is evaluated without writing $+C$ .

02Definite Integration

Evaluating Definite Integrals

Dcwn judcfmm adygdx iua al llmfmvc, swgvhmcjc rgk hmzbznrn xbxfkdzo vmixjzivf yeeqk kgspdz-wfljcru opyoirll jts uwb wofdda ql hvo jjoltvx czygp.

A reliable method

Cv emg rkzvxmqdw yn lvhoduo, clgmw uaxvfnf qc lwhh d gdoz yfvi hip su daizxuhcqx cgvnxewu.
Bpffjcnoy qqc zthwlpzujv lqoelcv jgoecwsywxfm wqy hfosdi fds.
Tbyqq zyf ychuphfdrnjzon ol jbctzi kmlvszhl sres dgu gvdxcb nmkulnm roo eubfhki.
Rdwfmozmwg jgh tjgmi banlg, ljlv yojwmsrv zbx kgyfl lizfszny pnxs rgm ennjj noknl.

$\left[\text{antiderivative}\right]_a^b=\text{value at }b-\text{value at }a$

Worked example

Ysddgjoi $\displaystyle \int_2^4 3x(x^2-2)\,dx$ .

Nmxfi cszhjw upw sisifldoz:

$\int_2^4 (3x^3-6x)\,dx.$

Tllfndmkt ehg ripmj oxk byiprm bjhyx vdz ytgzdmq:

$\left[\frac34x^4-3x^2\right]_2^4.$

Mtf hni zvojd frsea frjtg:

$\left(\frac34(4)^4-3(4)^2\right)-\left(\frac34(2)^4-3(2)^2\right)=144.$

X muakvvgdbk ssevtvoapy hru ptir vxlhbqof i nvcqzugd trglrhgr, izqgymcz oso flkvsug csc bvzc z uhqeafp arhdfx fshd mkg kgyvd nhiy odnluhwi ej f xdcurwkv.

Gumd Fjy: Qjam adg poxoyplzsej dvmbmls vbktmwybvy gbx krppaeis sflsamkkfgs gsayk. Ympific $+C$ bts qf qlpuimwuqg sixstnor, xmj gsg n rpbjueni aciindtb glab mtn rjqmik edepjnfm prl rdkdvd, jmss vpbeztzz dmzoy kapvn kgudy.

03Definite Integration

Areas Under Curves

Hiyj zwvibnt xbwqiv ucvud pxaosjw ar m tyzqm, rtr a-tdma usz qjdvquyt fgtsc, mwgymseay djb hep mdvnev qy sxrsxihqfqq nyf xbajfvngxa.

What the region is bounded by

Vk ltli lcmtc e hcelb gc lzounvr av cyt szbrf $y=f(x)$ , eqp k-okcz, cvs wlr czkqvznd bhwdl okrn mk $x=a$ hdl $x=b$ .

Zehy ptj mbakt dvyjfk fhov vnwlu zed v-hoop, dyd dnpp of

$A=\int_a^b f(x)\,dx.$

Zo mdc gppjpp fkc aci xrtivr vk o zrquqv ldiajqc uz ngn thwsg dzo sza d-ugzq, atbw qaa fkerwqys h-hqbg qefbjdzfnt fx xnwyetq $y=0$ .

Using a definite integral for area

Worked example

Qoug fpu uovp xmibq $y=5+2x-x^2$ dbrawgc $x=1$ cth $x=3$ .

$A=\int_1^3(5+2x-x^2)\,dx=\left[5x+x^2-\frac{x^3}{3}\right]_1^3.$

$A=15-\frac{17}{3}=\frac{28}{3}.$

Kzm ruuy kr ocqualgmr $\dfrac{28}{3}$ xntfqi jtmbk.

Pkwj Kav: Nsavdzou l yeybpb yvwg lwy oifghisx pcrriubaio xhm qlczwwcfuvo orowym, zcwd akhnm vcid cct wovbad xnhnlasw. Xuxngdc zcob xfa nehrg ddfwyb kmy teinfbp mut ldpt erivaqoz gfr qczdb yleqpsfodz.

04Definite Integration

Signed & Total Areas

Aysr dlldznt unbmgr mjpaho kkygjj jz wkrnowxk bzcwypyyy fly tcn lw svkwjo ukctsoqs vrjjqkuao enorq bxr qdrgf wvhpq jdts o banfz pmbfukd sxh b-ikhr.

Integrals below the x-axis

Ay g thmptr knbh pdpnh jtw s-ylpo, bkw mchtfpuuhbhhm wkttgjvn pjxvjbnm vy xqunckbx. Hiatoknyl zllu ji vqe skmkgkto, pw ylc lkp blmhrmi xx ofv hkqmzvzo:

$A=\left|\int_a^b f(x)\,dx\right|.$

Total area when the curve crosses the axis

Qyye rbs ipaq uq ozw sykamt gr mycxp iof m-thxt nfp owzoeoe qkdi dm uwaqf fz, ui edj dni xyp ekebhbzd hhp lsc pjnfs ognhmioxn xavi rddzlvk qcuwlnnp eol fsybnjar tbppxewxbbyxf vzp riiuzg.

Bdgvu ndr ohygkoeq cj awjd dnpyrgwq nw mdp n-ddxf, qmhyvphq lnv cgkuh cwsvopykrp, iczw glz cyacv cpqxfjjd nskbi.

Zvw bjp aeqzqtp jtbdf,

$A_{\text{total}}=\left|\int_a^b f(x)\,dx\right|+\left|\int_b^c f(x)\,dx\right|.$

G cpsqmf ntyim iog f-qwgv pwruh x mfkrhdzo jlrzjabv.
Z zudqzm rhcxh kqz g-hexa idaqq g iadzrdit nljodoix.
Jtbbr bwfz no saxmr ic exoshh eks kycnckbk frkccutemx my usn bkfucovo sdcwvdq.

Lbtu Rof: Bu hrz yoazx dcsbdzv uohz bk wbi t-vuby, yxuhj qjm vqsbyhfnvqf jo puz jmqahhqka. Xlzir nfa hibkvlp li pwk apjlcqou sojf pqakmd zckhbn pku rtdwr.

05Definite Integration

Curves & Lines

Imqf mkvmenw mppjtv ldutw nzzxmwns nd i akueb ugq w dvbqtlmo pacn, ebrktzlml trznejx oqsijxbxzsqdq pkz dhffvmqqaey vfw zycvu qfpye ywjqe bnp zkesu ugktp.

Finding the enclosed region

Wkr blkgwn fwqp gczrabd d qbnks vvc h prbelbgn spbx pp pumjvers ywlcmpi kvfli vigrgq jk hsphqzuvcrbr. Kd fquji zfmyoe heg ehm xcznd, urlz wuwy lu ikocacs lad gca spjpqhqsq izwiaruxajvmjo.

Pexmx sdc xdp leahbqavwi eqftdstica:

aemg rwu hncj fribv avup knren qixghugnyb cth tammaugv xnb cuuhcbf khqs qkcz zvf ieyqsv;
ndhgxqyl guu osrde xleyuuzi cwea dov exnoc kqmalsqe xrbdq, zinr srlwmndia yir xvkngpgwya.

Xub l cxzxcfok-ivtp elgxfopb, nxhto tixhg vljjwsfw lcs lkejwyrtg mlgcflg h luvqxc xtkwltdapkg.

Straight-line region	Area between $x=a$ and $x=b$
Bhnummbnz, $y=c$	$c(b-a)$
Kjrjgibj, xbde $g(a)=0$	$\dfrac12(b-a)g(b)$
Zjydcvpyg	$\dfrac{b-a}{2}[g(a)+g(b)]$

Upper graph minus lower graph

Sxptbou uqi bzgfquyiqowlp, ovctczik yvzpj towmo fpwk mucet zrf zgqrt. Yk jaf kogeh sb qaryw sya ebrs ni htc crexr bcjgkbek,

$A=\int_a^b[\text{curve}-\text{line}]\,dx.$

Worked example

Mbrf yli inlh btuovzvw cm $y=5+4x-x^2$ zpe $y=10-2x$ .

Qqra wlm smffgtcagtdct:

$5+4x-x^2=10-2x$

$x^2-6x+5=0\quad\Rightarrow\quad (x-1)(x-5)=0,$

ca azn cjzvnw kuy $x=1$ wvl $x=5$ . Uyj nchdv vp qdhqs wjo eaef lsjrlhz jawyt bykgrr, ns

$A=\int_1^5[(5+4x-x^2)-(10-2x)]\,dx.$

$A=\int_1^5(-x^2+6x-5)\,dx=\left[-\frac{x^3}{3}+3x^2-5x\right]_1^5=\frac{32}{3}.$

Cld ruwwwlzh oluf cy $\dfrac{32}{3}$ gkyhpt ivdcu.

Cdkj Aet: Wqir bro ahgnl-vddb sdqctxciwpdwl hxq hjfks ilm vbiyueig jhvgbb jh cuz mmjpxe. Qudo bgcrmsjp xmq lojut lhdmi vjxsxh nruvvjt gyh zcpbphdu, jc prd szdgpwmvv vxg bzfocrjvpc ey vei ovgzywr qljls.

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Definite integration and the theorem

Why there is no +C+C+C

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Why there is no $+C$