University:
Fresno Pacific University
Course:
MATH 210 | Calculus I
Academic year:

2024
Views:

375

Pages:

4
Author:

Lily Callahan

Name Period Honors Brief Calc. 15.5 Notes Finding Areas Integrals were first discovered while solving a geometry problem : Given a nonnegative function f, whose domain is the closed interval [a,b], what is the area enclosed by the graph of f, the xaxis, and the vertical lines x =a and x=b? SB u f Area under graph of f from a to b = Ja if 1. I f f is continuous [a, b] f ( x )d x 2. f is nonnegative [a,b], that is f(x) 0 for a x b 3. An antiderivative for f can be found Properties of area: Nonnegative Property: area > 0 I Additive Property: If A and B are two nonoverlapping regions with areas that are II. known, then Total Area of A and B = Area of A + Area of B . - EX 1: Find the area under the graph of f i x ) = x 2 f r o m x = 0 t o x 5, * 4]c y = x2 / If f i x ) < 0 on a x =1 b, then — f {x) t a i > : 52 • (3 0, and b y symmetry , the area A equals - f ( x )d x = - I I Ja f ( x )d x J- L < , .. V ,7 —— I - - kX 3 l' • Jutf M 'V' £X ' In . = -1, and x = -2 f ( x ) z C 0.o, i EX 2: Find the area under the graph of / (x) = x3 , the x - axis x qrora - -L 4- — j ~ i - H4 . -4 1 5 r -I .- . — »1 1 4 r ' f- —+ J +M r i £ -t » *• i • % ~ -4 J- -h -f L J. . 1 * •i I 1 i % 3 -+ 1 ] M3 - 0 i- A= H EX 3: Findthe area under the graph of / (x) m tv 1i • V - i 4 ri r 5 —±t 1 £4 = I -- T“ i ““ / " 0 L + RT — — p r : : : . [ -~H + ailj--L -U t "* 1 4 * --!rr -r 4.1 i ‘r 4 SI - -- rhife- , 32 L irXt - V * 4 1 4.1r ^ "" r * ‘ JU^ i 4. ‘t . .?f . rH - ' a - *3 3 7 xl-i - i ¥ 3 4 x 30 • k* — rx 4 L d- * IA rl 'y 3 • > 2s \ iVl -|v? MfP _ Jxtii • - — - :; 1 rr r- p -J . L. X. - 3c3 3 4-H rr H__ LX-wAjrrr-4-UVh - ..T 14 3 r r — — - t Pf -V'.X f r - -f ; EE: -- - . a J, rco -. i • ' - 2- "'! ' —— i ' . 1 i ~J i . i ! -X / X- C, U i T-f — —\ - ci 1 ' 3L < ar f V t ) v trtv > " r4 + ? ) - ( i V I A 1 - 0 < 3 4 It Sx 4- f + ELJ . y “ Cx ") - r Ui is - o.rpu 4 xR ^ 2X 4- X- 1 ft 4 1I ** i J:: XJX : = 2x + 4 ? X 3 ? y . i f - W/ > ;y ( y 5 ... X ) :r 0 ^ • T 1 J y - 4 t1 i 3 > -2 a. ^ zX o . 13 { Ifea - i \ ; EX 4: Find the area enclosed by the graphs of the functions f ( x ) = 2 x 2 a n d q ( x) ' L? Li H, 3 1 *1 i H I H t , \c 1 fo cr0- H 1 L L ( 1 ,J Frr - - - 4 fi "1 V *4 ^ — axis from x = 0 to x = 4 4 and the x 0 ya I 1 I \ J £ *] J- <« V J 1 4 )- ' } I i i r ~ 1 — - "i 0 l, /- __ n - j -r - -u -44 ~4 — 7- ; rt x 2 1 C? f+ M " iXtxi:pi 4:~ . ' ..4 ,. j ' 1 ‘ -r fU xTmxixi - -RK - W t .T I X-u,- / ~ *H1 X * i:. 4 f imx . 0 * -' v . • • 1 L ) X J J Tf . - / A . > = v. / • -bp liit oh \ y( fo ' — — J Y * . Ay yyw ^X - 3 { 3X _ s : 4 i. 1 i .rt 1 « I . i ? . 3 (3, bO O ) 6 = - X 4 13x2 - 3ox r 1 I r J~ ~ -4 "» X i rt « f »- 4 t t •• l x- -— t- »« ~4 ~, — * ch J° 0 lo 9 a- 3 /0 y ~ y* -3ft 4 3y~ 3 ?-3ooV | 3 Aw C? EX 6: Find the area enclosed by the graphs of the functions ( .( ( / 0) = x 3 and g ( x) = -x 2 + 2 x ( iV3 4 tHO- f - * i . /4 f 0 IA 1* ' 'T XTi 4. xa < if y( y 3 4 ax. x -~ so [ 0 0 ) Co " £) A= fid f i fj * yV _ Ti... = -y ° ^ ju < 4 5/4 * /0 c*X f V) ^ How do you fine area between curves? I I I lSt I I Graph the curves i nd 2 Represent area element t6~ r + f(x )=x 2V I I I .1 i / / ai* Jx )=?x (T • 3rd Determine the dimension h=g(x)-f(x) and w=dx in this case. 0.5 ' i i V t » I V -0.5 Zla Ji! I 0 1.5 t a b 5 th, Accumulate the area using [&(*) - / MW* Ja 4th Find beginning “a” and end “b”. f j x 2 /x Graphic Organizer by Dale Graham and Linda Meyer Thomas County Central High School Thomasville , Ga .