dx ¥1 i¥iiimi¥±¥y - github pages · lecture:5-1areas anddistances areas - the big question:...
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LECTURE: 5-1 AREAS AND DISTANCES
Areas - The Big Question: How Might you Find Area Under a Curvy Curve?
Example 1: Divide the interval [0, 1] into n = 4 sub-intervals of equal width. Then, use four rectangles to estimatethe area under y = x2 from 0 to 1.
(a) Using left endpoints.
(b) Using right endpoints.
UAF Calculus I 1 5-1 Areas and Distances
^ into n pieces l sub intervals )
µf 8294¥kraekaaahbnkggeofeawsub . interval
- �3� make it belter ? Make more
rectangles !1
1 1 1 1 1 i )a b
Width of sub- intervals is Dx = but = 1¥ = ya,
^L4= ty ( at ty fCY4 ) + tyf(Yz)ttyfC3/4)
¥1, I¥iIImI¥±¥Y"Yz by ! = LT . Yg
= 7132=0*21875
×134 =ttflY4 ) + taftk) + tyf (3/4) + tyfu )
= ty ( ( Yap t ( Yzp +1314ft 12)= ty ( 416 t 4444
t 946 +the ).¥
k by i) = ty ( 3£ )
= 1513220.468£
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To find the actual area we need to take the number of sub-intervals to . To do this we need ageneral expression for the left or right estimate for any n. This process is rather tedious and we will soon learnhow we can use Calculus to find area under curves without having to use this long, tedious process.
Example 2: Prove that the area under y = x2 from 0 to 1 is 13 .
UAF Calculus I 2 5-1 Areas and Distances
infinity
Rn= In ( fl Yn) + fl4n ) tfpln ) + . . . + f( %) )
=tnn÷+2÷+n3÷+ ... + In. )⇒athtyn . . . In
=
nzt ( 12+22+34 . . . + n2 )__
this adds upto nlhtD_tD
6= ¥ nlt )
6
=
(nti)l2ntD6n2
= Znhtnttntl.2
=
2nhy.gr# / thisgives the right sum
or any n
To get the exact area we take n ( #of sub . intervals )
to a .
A = lningn Rn
= lim (ftp.t#)Ynznsn /n2
=
hiya( stem )
= @ ← area under y=x2 on [ 0 , Dis exactly 43 .
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Upper and Lower Sums: In general, we form lower (and upper) sums by choosing the sample points x⇤i sothat f(x⇤i ) is the minimum (and maximum) value of f on the ith sub-interval.
Example 3: Estimate the area under f(x) = 2 + x2, [�2, 2] with n = 4 using
(a) Upper Sums (b) Lower Sums
Question: What type of behavior will guarantee that the left sum is an under-estimate and the right sum is anover-estimate?
Example 4: Find an expression for the area under the graph of f(x) =px, 1 x 16 as a limit. Do NOT evaluate
the limit.
UAF Calculus I 3 5-1 Areas and Distances
Bit 2I¥=±
"*:# ¥ •II l l l ) # ]I I I I ?Vµ=|lfH)tffDtf(1) + HZD Lower -- 1.( ftp.fldtfldtfa ) )= (6+3+3+6) =3+2+2+3=18
= 10
Increase decrease ? concave up/ concave down ?
¥.
r \ ,'a b 'a to
'a b
is increasing Ln is under - est , Rn is over - est
If f is decreasing Ln is ovekest , Rn is under . est
D×=Ent=In Rn=
his. (
lfthtntttnstfffsat. . . F
#=i¥FMi¥
my )
• '
l' . a
.
i
A= "nYgRn=nh;g§g' EVENA HEN Hunts' H3l¥ )
16=1+
n (In)
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Example 5: Determine a region whose area is equal to the given limit.
(a) limn!1
nX
i=1
2
n
✓5 +
2i
n
◆10 (b) limn!1
nX
i=1
5
n
sin
✓2 +
5i
n
◆
Example 6:
(a) Use six rectangles to find estimates of each type forthe area under the given graph of f from x = 0 tox = 12.
(i) L6
(ii) R6
(iii) M6
(b) Is L6 an underestimate or overestimate of the truearea? Is R6 an underestimate or overestimate of thetrue area?
(c) Which of the numbers L6, R6 or M6 gives the bestestimate? Explain.
UAF Calculus I 4 5-1 Areas and Distances
yaxiniewivdatnotfstnprtfihnf
fDX= width of interval- starting point .
This is the area under This isthe area under
fcx ) = x'0
on [ 5,7 ] . f ( × ) = sinx on [ 2,7 ]
.8
~~2( 9+8.8+8.2 -17.2+6+4 ):
=86@=2( 8.8+8.2+7 . 2+6+4+1 )
=7O@A I
1=218.9+8.5+7.7+6.5+5+3 ) 2 6 10
= 799€
1-6 is an overestimate
126 is an underestimate
Mb appears to be best as it's
over - estimates seem to cancel out
w/ underestimates .
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Distances
Example 7: Oil leaked out of a tank at a rate of r(t) liters per hour. The rate decreased as time passed and valuesof the rate at 2 hour time intervals are shown in the table. Find lower and upper estimates for the total amount ofoil that leaked out.
t (h) 0 2 4 6 8 10r(t) (L/h) 8.7 7.6 6.8 6.2 5.7 5.3
Example 8: Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30second time interval. We take speedometer readings every five seconds and then record them in the table below.Estimate the distance traveled by the car using a left sum and a right sum.
Time (s) 0 5 10 15 20 25 30Velocity (mi/h) 17 21 24 29 32 31 28
UAF Calculus I 5 5-1 Areas and Distances
If velocity is constant dist = vel * time
[email protected] = mi
rank) units Yhr
¥µupper est is ↳
L5=2( 8.7+7.646.8+6.2+5.7 )¥A#[email protected]+6.8+6.2+5.7+5.3 )= 63€
zsotu ywatrnneihbstamnetunibinner
.int#utrFminitehEec=5h0=zEg8o=2oolo4=tf3o2=6q6g7*l24n93LokselEpdkm1u@+gePhVErtmYnmft1sec
mumLb = 5( 24.933+30.8 +35.2+42.533+46.933+45.467 )
= 1,129.33£Rb= 5130.8+35.2+42.533+46.933 +45.467+41.067 )
= 1210€