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1. [-/2.22 Points]
MY NOTES
SCALC9 4.1.001.
DETAILS
11
ASK YOUR TEACH ER
11
PRACTI CE ANOTI-iER
U n less specified , all approximating recta ngles are assumed to have the same w idth.
T he graph of the function f Is g iven.
y
+
6
y
= ftx)
4
2
+-------L-- -----'-- - - - - x
4
(a)
8
Ci)
By reading values from the g iven graph off, use five rectangles to find a lower estimate for the area under the given graph off, the x-axis, and the lines x
to o ne deci mal place.)
Sketch the rectangles to Illustrate the estimate.
=0
and x
= 10.
( Round you r answer
~
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4 .1 HW: Areas and Distanc e -
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=0
and x
= 10 .
( Ro u nd your answe r
te.
Sketch the rectan gles to Illust rate the estima
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y
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x -axis, and the lines x
for the area und er the g iven graph of f, the
use flve rectan gles to flnd a lower estima te
By readin g values from t he given graph off,
to one decim al p lace.)
y
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y
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y = fi x )
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l
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4
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By read ing v alues f rom the given graph of f , use five rectangles to find an upper estim ate for the area under t he g iven graph of f, the x-axis, and the ll nes x
to one decimal p lace. )
Sketch the rectangles to Illustrate the estimat e.
y
y
y
6
6
y = fix)
4
y = fi x )
y
4
= ft x)
=0
and x
= 10. ( Rou nd your answer
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() Resources for 4.1: Fal21 MATH x / ( : 4.1 HW: Areas and Distance
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II-Ji. 4.1 HW: Areas and Distance - I
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webassign.net/w eb/Student/Assignment-Res p onses/ las t?dep= 27 003985
By read ing valu es f rom the given gra p h off, use five rectang les t o fi nd an u pper e st imate for the area under the given graph of f, t he x - axis, and the lin es x • 0 and x • 10. ( Round your an sw er
t o one d ecim al p lace.)
Sketch t h e rectangl es t o Illustrate the est imate.
y
,-
y
y
6
y
= fix)
4
2
0
y
8
X
X
X
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X
4
0
y
+
8
X
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~-t----+----t--
1
0
( b)
+
4
8
Ci)
Find new estimates using ten rectangles in each case. (Round your answers to one decimal place. )
lower estimate
upper estimate
8
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() Resources for 4.1: Fal21 MAT!-<
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4.1 HW. Areas and Distance
V-,.. d .1 HW: Areas and Distance -
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2. [-/2.22 Points]
DETAILS
SCALC9 4.1 .002.
M Y N O TE S
Th e graph of the fu nction f Is given.
y
+
i
32
24
y
= ft. x )
16
8
4-0- 0- -+---8- - -l-6- -2-4- -3-2- -
- 4-8-
X
Ci)
( a)
Use six rectangles to fi nd estimates of each type for the a rea under t he g iv e n graph o f f, t h e x -ax is, a nd the lines x
(I)
L 6 ( sample points are l eft e ndpoints)
L6 =1~ - - - ~
(II)
R 6 (sample points are right e ndpoints)
R6 = ~i_ _ __
(111) M 6 (sample points are midpoints)
M6
=
=o
and x ~ 1 1.
11
ASK YOUR TE ACHER
J [ ! "RAC rl C[ J\l'l
• • •
0
Resources for 4.1: Fal21 MATH x
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4.1 HW: Areas and Distance
x
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4.1 HW: Areas and Distance - F
x
+
webassign.net/web/Student/Assignment-Responses/last?dep=27003985
(a) Use six rectangles to find estimates of each type for the area under the given graph off, the x-axis, and the lines x
(1)
L
6
O
O and x "' 48.
(sample points are leh endpoints)
L6 ""'~'- - ~
(Ii)
R
6
(sample points are right endpoints)
R6= ~'- - ~
(ill) M 6 (sample points are midpoints)
M6"' ~'- - ~
(b)
Is L 6 an underestimate or overestimate of the true area?
0
0
(c)
(d)
Is R
overestimate
underestimate
6
an underest,l mate or overest_lmate of the true area?
0
overestimate
0
underesti mate
Which of the numbers L , R 6 , or M 6 gives the best estimate of the true area ?
6
Q
L6
Q
M6
Q
R6
3. [-/2.22 Po ints]
DETAILS
;:;iiiiiii.~~
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I
SCALC9 4.1.504.XP .MI.
a~ rel~
si 0
[ MY NOTES
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4.1 HW: Areas and Distance -
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3. [- /2.22 Points]
DETAILS
=I
*
SCALC9 4.1 .504.XP.MI.
(a) Estimate t he area und e r the graph of f(x)
R4
x
4.1 HW: Areas and Distance
I
= 10
MY NOTE S
cos(x) from
x
= Oto x = ,r/2 using four approximating
y
ftx) =
10 C0 S(X)
ftX)
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
7f
11
3 7f
7f
8
4
8
2
y
; --
X
0
- - ~ ~11
8
---11~-
-
= 10 C0S( X)
-
4
-::3 ~,r- - - -,r
"- X
8
2
y
/(X)
= 10 CO S(X)
10
9
8
ASK YOUR TEACHER
I\
PRACTICE AN OTHER
rectangles and right e ndpoints. (Round your answ ers to four decimal p lace s .)
Sketch t he graph and t h e r ectangles.
10
I\
/( X)
= 10
C0 S( X)
I z@.
0
➔
C
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I () 4.1 HW: Areas and Distance
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y
y
10
ftx)
10
ftx)
+
= 10 cos(x )
= 10 cos(x)
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
,r
,r
3 ,r
,r
8
4
8
2
X
,r
0
~8
,r
3 ,r
4
8
Is your estimate an underestim ate or an overestim a te?
0
underestim ate
0
overestima te
(b) Repeat part (a) u sing left endpoints.
L4 =
I
I
Sketch the graph and the rectangles.
y
y
10
9
8
ftx)
= 10 COS ( X )
ft X) =
10 COS( X)
;r X
2
• • •
Q
Resources for 4.1: Fal21 MATH
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I () 4.1 HW: Area s and Distance
webassign.net/web/Student/Assign
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(b) Rep eat part (a) usin g left
end poin ts.
L4
=
I
I
Ske tch the grap h and the rect
ang les.
y
y
10
f{X)
10
= 10 COS ( X)
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
9
8
7
6
= 10 COS( X)
~,r
,r
3,r
Tr
8
4
8
2
X
0
y
10
f{X )
,r
,r
3 ,r
Fr
8
4
8
2
y
JlX)
= 10 COS( X)
10
9
8
7
6
/(X)
= 10 COS( X)
X
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4.1 HW: Areas and Distance
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0
1l
1l
37f
8
4
8
\
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X
1l
2
0
y
3 ,r
,r
8
4
8
2
10
ft..x) = 10 COS( X)
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
1l
1l
3 ,r
,r
8
4
8
2
Is your estimate an underesti mate or an overestimate?
0 underestimate
0 overestimate
,r
X
y
10
0
r.
*
X
0
ft..X) = 10 COS( X)
~-
,r
,r
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4.1 HW: Areas and Distance
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4.1 HW: Areas and Distance - f
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webassign.net/web/Stud ent/Assignment- Responses/last?dep= 27003985
i
4. [-/2.22 Points]
Let f (x)
(a)
x
=6
DETAILS
MY NOTES
SCALC9 4.1 .005.
+ 4x 2 .
Estimate the area under the graph off, the x - axis, and the lines x
R3 =
11 ASK YOUR TEACHER 11 PRACTI CE ANOTH ER
= -1 and x = 2 using three rectangles and a right endpoints.
IL___ _,
Improve your est imat e by using six rectangles.
R5
=
L I_ _ ___,
Sketch the cu rve and t he approximating rectangles for R 3 .
y
y
y
20
20
20
15
15
15
10 1-- - - --f
10
·- - - -- -
5
5
5
)I
2
X
X
X
1
(j) 0 -1
1
2
2
(j)
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Sketch the curve and t he approxima ting rectangles for R 6 .
y
y
y
20
15
I
I
10
I
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(b)
Repeat part (a) using a le~ endpoints.
L3 =
'l...--------'
Sk etch the curve and the approximat ing rectang les for L 3 .
y
y
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4.1 HW: Areas and Distance -
r x
Sat 7:10:27 PM
(b)
Repeat part ( a) using a le~ e ndpoints.
L3 = L.I_ __
...J
L5 = L.I_ _ __,
Sketch t he curve and the approximating rectangles for L 3 .
y
y
y
5
0-1
y
2
X
X
X
1
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1
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1
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(i)
X
X
o-1
1
2
1
(j) 0- 1
y
X
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1
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Sketch the curve and the approximating rectangles for L 6 .
y
20
20
15
a ~ ..
y
y
20
2
(j)
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6
y
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20
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Repeat part (a) usin g a m idpoi nts.
M3
=
M6
=
' - I- - - - - - -
' - I- - - - - - -
Sketch the curve and the approximating rectang les fo r M .
3
--
X
- 0.5
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1
1.5
2
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1~ 1·
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Repeat part (a) using a m idpoints.
-I
::=======:::
M6 •~'----
Sketch the curve and the approximating rectangles for M 3 .
y
y
y
20
15
o- 1
y
2
X
X
X
1
G) 0 -1
1
0 ·0
+
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M3
Q.
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2
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0 0- 1
y
0-1
Sketch the curve and the approxima ting rectangles for M 6 .
y
y
y
2
(j)
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Sketch the curve and the approximating rectangles for M 6 .
I
y
y
y
I
X
o- 1
0.5
-0.5
y
1
1.5
2
(i) 0 - 1
-0.5
0.5
1
1.5
0 ·0
+
webassign.net/web/Student/Assignm ent- Responses/last?dep=27003985
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X
0-1
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From you r sketches In parts (a) - (c), which estimate appears to be the most accurate?
Q
M
Q
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L6
6
6
1
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5. [-/2.22 Points)
I
MY NOTES
SCALC9 4. 1.016.
DETAILS
h
Use this definition with right endpoints to find an expression for the area under the grap
ftx)
6. [-/2.22 Points)
=~
9x
x-+ 7
,
Of f
I[
ASK YOUR TE ACHER
as a llmlt • Do not evaluate the llmlt.
1:Sx:S3
MY NO TE S
SCALC9 4.1.017.
DETAILS
11 ASK YOUR TEACHER
the limit.
use thls definition with right endpoints fo find an expression for the area under the graph off as a limit. Do not evaluate
2
ftx) = 9 + sln (x),
7. [- /2.22 Points]
0 :S x :S n
DETAIL S
SCALC9 4.1.018.
~
MY NOTES
the llmlt.
Use this definit ion with right endpoints to find an express ion for the a rea under the graph of f as a limit. Do not evaluate
f(x)
= x2 +
~,
7
s
x
s
9
11 ASK YOUR TEACHER
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4.1 HW: Areas and Distance
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4.1 HW: Areas and Distance - F x
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DETAILS
SCALC9 4.1 .018.
0 ,_
+
0
webassign.net/web/ Student/Assignment-Responses/last?dep=27003985
7. (- /2.2 2 Points]
Q.
* • CD
M Y NOTES
11
ASK YOUR TEACHER
11
PRACTI CE ANOTH ER
Use this definition with right endpoints to flnd an expression for the area under the g raph off as a lim it. D o not eva luat e the limit .
f(x)
1
= x2 +
8. (-/2.22 Poin ts]
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,
DETAILS
7 s x s 9
SCALC9 4.1 .020.
M Y N OTES
11
ASK YOUR TE ACH ER
Det er mine a region whose area Is equal t o the g i ven llm lt . Do not evaluate t he limit.
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8
llm
\
n-00
0 x9
0
~
I = 1
n
n
on ( 1 , 2)
x 9 on [0, 1)
0 x 8 on
0 x
7
(0, 1)
on (0, 1)
0 x 8 on
(1, 2)
- - - - - - --
- - - . . . . j . _ ~ - ~ - - - - - - - - - -........-- - - - --
- : : - - - - - - - - --=---"'M-Y_N_o_TE_s__ [
ASK YOUR TEACHER
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I\
PRACTI CE ANOTHER
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