Calculus 2 labs
, Lab 1 : Calc 1 review
evaluate derivatives
]
"
b)[sin()]5 이음[ ( ( tx ) α
ex
a)(Sx" -
x4)) S(sinx( (cos(x))
* (1 + xz)3xe
+
(2x)ex3
+
(5x3 xy)6/1Sx 4xS)
?
< -
-
01 前[e ] ('+ t)なと 外
2(6x) eH
" .
=> = (3x2 1) +
evaluate using U-Sub
a)((xi + x)5(3x2 + 1)dx 6)USsech(5x + 1)dx Stanx dx
c)
U =
3
X tx du =
3x+ 18x
u = Sx + 1 du = 5dX δor
Usec du du cosxdt
Judu =
v
=
Sinx =
u
+
.√ st
~ Fanut (
= 踏 c
tan(Sx + 1) +C
3
a) *
dx
더
e
0x =
=13 du =bxdX
eeduc
,Lab2 : Area Between curves
Area
= [f(x) g(x)] -
dy
. Find the
1 area bounded by x
Mxty =
y and
=
range (-5 ,
1)
gaxe and
5 4x
y
%
-
=
2
xi * + Mx 5 0 .
2 cost
y
=
=
-
g =
:
,
10 T if x = /4
m + 5 if X =
0 > 2 =
cos/Ty4)
=
y (x + 5)(x 1) ,
=0 Solm.
線
-
itt = 8 f ( 5) y =
and
y
=0
, i 0
= or
y
cTB (cosx 0]
a
y 5 4x
A g
= -
店
室¤
x2 = 5 -
Hx 5 = -
5 .
1 = -
A
=[s - nx -
x2]dx = A =
(sx 2x2 -
-
13) =
Whe cosix
Sinxl .*
[S6) 2112 -
:崎)S65 )
-26- 5
㎡] =
(8 _
2
-5
で - 505
答) f 3
ふ 宮 だ35 +警] =
ら
、 “ 法路答
~ 8-
52= 36
= (f(y) g(y)]dy
78 撃
A -
A =
r ( = *Ey
-
iy] ? _
Cine- * 2]- (lm-i)]
)
= 0 + 市
Eo(o-X]d
:
} . ) ×次 )+ 통다
ytl iftar
=
" e ,
# (x
#
x o)dx
. ,
い
-
-
節磁機築出 ·
σ
↑=砦
“
琵治 ↑ rx =武
器
ndo =
主信 。 ”
) 。
0 。 に
}
予
( 9 %。
} =) 算
A 臨訾
言
一
, Lab 3 : Disc and Washer's Method
Disc : find the volume of the
1
.
solid
generated by revolving the shaded region
about the X-axis
Tt ay C
Soln
=
:
品 [R [ }} oi
'
Va
y 響
E
]
X+ 2 と = て
v
=π^ [ … " 是
0
- 0r
0
=℃[ -
?
) += ( に) ( 1
ω - 些)
ㅇ X
V =π
- [ + 北
' ≈
き
]
2
v
=π㎡ / 6 ~+ 背) of
品”
(
くに
制“
π
“
了
。2^
2
SoNIb [R(y)] dy
3 .
φ=㎡ , Y
た
0 ×= 2
soln :
v = [tany-o] "dy =#22)"OA
蔀嶽
언
=
せ ‰ ' ta ? (y ay
=Th2 + Of
포y
ton( Sechy-tan(y) =
1
=π
(음]:
= Ean ( ]= ScE 2 { } ~ !
# " sec - % 号
π[ 崎 了
v=
v
に
-響 .>
v =π
ㅇ ] v
*( 倍tany}!
5
=
=π
[晋 tam [3- } [tan0 -
歌 )- iY3 : (
-). } 0
=π 浩
いに 4 - π
washers method : inner radius
v=
[R(x) -
r(x)
=
]dx
outerradius Soln !
“ に- Y +
니 φ 며
yirasz on
Cost
2
·
v =競 ( cos + ) dr
芝
-
+
플
T2
v =
π(X -
sinx)
Tyz
sint 乳) )
-
v π ( -
sin ( 乳) ~
( -
v= (乳 -
1 ) 事 1) 2
0 :
π ( 1 +π+ 1 ) =π _
π
己一一
一
二
長
一長
一
乙
, Lab 1 : Calc 1 review
evaluate derivatives
]
"
b)[sin()]5 이음[ ( ( tx ) α
ex
a)(Sx" -
x4)) S(sinx( (cos(x))
* (1 + xz)3xe
+
(2x)ex3
+
(5x3 xy)6/1Sx 4xS)
?
< -
-
01 前[e ] ('+ t)なと 外
2(6x) eH
" .
=> = (3x2 1) +
evaluate using U-Sub
a)((xi + x)5(3x2 + 1)dx 6)USsech(5x + 1)dx Stanx dx
c)
U =
3
X tx du =
3x+ 18x
u = Sx + 1 du = 5dX δor
Usec du du cosxdt
Judu =
v
=
Sinx =
u
+
.√ st
~ Fanut (
= 踏 c
tan(Sx + 1) +C
3
a) *
dx
더
e
0x =
=13 du =bxdX
eeduc
,Lab2 : Area Between curves
Area
= [f(x) g(x)] -
dy
. Find the
1 area bounded by x
Mxty =
y and
=
range (-5 ,
1)
gaxe and
5 4x
y
%
-
=
2
xi * + Mx 5 0 .
2 cost
y
=
=
-
g =
:
,
10 T if x = /4
m + 5 if X =
0 > 2 =
cos/Ty4)
=
y (x + 5)(x 1) ,
=0 Solm.
線
-
itt = 8 f ( 5) y =
and
y
=0
, i 0
= or
y
cTB (cosx 0]
a
y 5 4x
A g
= -
店
室¤
x2 = 5 -
Hx 5 = -
5 .
1 = -
A
=[s - nx -
x2]dx = A =
(sx 2x2 -
-
13) =
Whe cosix
Sinxl .*
[S6) 2112 -
:崎)S65 )
-26- 5
㎡] =
(8 _
2
-5
で - 505
答) f 3
ふ 宮 だ35 +警] =
ら
、 “ 法路答
~ 8-
52= 36
= (f(y) g(y)]dy
78 撃
A -
A =
r ( = *Ey
-
iy] ? _
Cine- * 2]- (lm-i)]
)
= 0 + 市
Eo(o-X]d
:
} . ) ×次 )+ 통다
ytl iftar
=
" e ,
# (x
#
x o)dx
. ,
い
-
-
節磁機築出 ·
σ
↑=砦
“
琵治 ↑ rx =武
器
ndo =
主信 。 ”
) 。
0 。 に
}
予
( 9 %。
} =) 算
A 臨訾
言
一
, Lab 3 : Disc and Washer's Method
Disc : find the volume of the
1
.
solid
generated by revolving the shaded region
about the X-axis
Tt ay C
Soln
=
:
品 [R [ }} oi
'
Va
y 響
E
]
X+ 2 と = て
v
=π^ [ … " 是
0
- 0r
0
=℃[ -
?
) += ( に) ( 1
ω - 些)
ㅇ X
V =π
- [ + 北
' ≈
き
]
2
v
=π㎡ / 6 ~+ 背) of
品”
(
くに
制“
π
“
了
。2^
2
SoNIb [R(y)] dy
3 .
φ=㎡ , Y
た
0 ×= 2
soln :
v = [tany-o] "dy =#22)"OA
蔀嶽
언
=
せ ‰ ' ta ? (y ay
=Th2 + Of
포y
ton( Sechy-tan(y) =
1
=π
(음]:
= Ean ( ]= ScE 2 { } ~ !
# " sec - % 号
π[ 崎 了
v=
v
に
-響 .>
v =π
ㅇ ] v
*( 倍tany}!
5
=
=π
[晋 tam [3- } [tan0 -
歌 )- iY3 : (
-). } 0
=π 浩
いに 4 - π
washers method : inner radius
v=
[R(x) -
r(x)
=
]dx
outerradius Soln !
“ に- Y +
니 φ 며
yirasz on
Cost
2
·
v =競 ( cos + ) dr
芝
-
+
플
T2
v =
π(X -
sinx)
Tyz
sint 乳) )
-
v π ( -
sin ( 乳) ~
( -
v= (乳 -
1 ) 事 1) 2
0 :
π ( 1 +π+ 1 ) =π _
π
己一一
一
二
長
一長
一
乙
