Version 2 - Integration

Class XI PHYSICS Class Assignment Integration

TOPIC:

RULES OF INTEGRATION 1 Evaluate the following indenite integrals

∫ 2 tdt 

a

∫ √  y y dy

!

"

∫

2

∫ (5 x +3 x −2 )dx 2

#

 dt  3 t 

$EFINITE INTEGRATIONS % Evaluate the following denite integrals π / 4

a

!

5

∫ (5 x + 3 x )dx 4

∫ cosxdx

∫ sinxdx

"

10

∫ ( x + sinx ) dx

#

∫  xdx

e

π / 6

0

0

2

π  /  / 3

π 

4

1

2

∫

&

1

9

9

dx 2  x

g

∫ y

1 /2

∫

'

dy

0

0

4

dy  y √  y

1

∫ ( x x − x ) dx

i

1

∫e

 x

 (

1

dx

0

)ETHO$ OF SU*STITUTION + Do the following integrations 7

a

7

1

∫ x +5 dx

7

1

∫ 4 x +5 dx

!

4

∫

"

4

4

 x  dx  x + 5

5

#

 x −2  dx + 5  x −2

∫

2

∫ (6 x +2 ) dx 3

e

−1 3

π / 3

&

∫

g

2

∫ (2 t − 4 ) dt 

sin ( 2 x ) dx

∫ ( x x +2 )  dx

+ 5)

 dx 3

∫

 (

 x

2

 x e dx

0

3

0

1/ 2

∫e

2 x

dx

0

l

e

m

π / 2

∫ sin ( x )cos ( x ) dx

∫ 1

ln

( x )

 x

dx

n

π / 3

∫ tanθdθ

π / 6

a

.

∫ 0

∫ cos θsinθdθ 2

π / 2 √ π 

∫ 2cos ( x ) xdx 2

π / 3

r

 xdx

0

1

3

2 2

( x x + a ) 2

o

π / 2

0

-

−3 2 x

1

2

1

∫(

i

2

π / 6

1

,

−2

'

∫ 0

s

2 xdx

∫ 0

3 x

5

dx 3/ 2

( x + 1 ) 3

∞

t

∫ e− 0

/ ( x  x + 16 ) 2

3 2

INTEGRATION AS SU))ATION OF S)ALL ELE)ENTS / Find the area bound by the curve y =  x 2 , x-axis and the line x = 3. 0 Find the area bound by the curve

− x

e

, x-axis and y-axis.

 Find the area enclosed by the curve  y = sinx  for x between a. 0 to  b. 0 to ! c. 0 to "! 2 # cube of side $a% &as shown in the gure' has (ass density given as  z  ρ= ρo ( 1 + )  where $a% is a constant. Find the (ass of the cube. a r adius ) and height * using integration. 3 Find the volu(e of a unifor( solid cylinder of radius

4 Find the volu(e of a unifor( solid cone &of radius ) and height *' using integration.

2 x

dx

15

 +he current assing through a cross-section as a function of ti(e is given by

i = i o t ( 1 + t )  Ampere  where $t% is in seconds. Find the total charge assed in the interval $0% to $t% seconds. *eat energy or ther(al energy &+E' contained in a body is given by TE= msT  , where 11 ( is the (ass, s is the secic heat and + is the te(erature of the body in elvin. # unifor( steel rod of length $% and secic heat $s% has 2

te(erature &+' as a function of x given by

 x T =T o (1+ 2 )  elvin. Find  L

the total ther(al energy of the rod. /otential energy of a oint (ass is given as $(gh% where ( is the (ass of the 1% article and $h% is its height fro( the ground and $g% is acceleration due to gravity. Find the otential energy of a unifor( rod of (ass $(% and length $l% that is standing vertically fro( the ground.

1+  +run of a tree is softer inside and harder outside. # articular iece of a wood trun &cylindrical in shae' has a radius $)% and thicness $t% has a density as a function of distance fro( the center as

 ρ= ρo ( 1− cr )  where $c%

is a constant. Find the (ass of the wood. Find the (ass of the wood in ter(s of and ).

 ρo , c, t

A""eleration6 7elo"it86 -osition an# #is-la"ement 1/  +he seed of a article travelling along 1x-direction is given as 2 = 3&1!t' ("s where $t% is ti(e in seconds. Find the dislace(ent of the article in the ti(e interval between t = ! to t = 4 s. # (otorcyclist who is (oving along an x-axis directed towards the east has an 10 acceleration given by a =6−1.2 t   c("s!. 5ts velocity and osition at ti(e t = 0 is given by v = ! ("s and x = 6.00 c(. Find its velocity, dislace(ent and osition as a function of  ti(e. 7hat is its osition after ti(e 4 s8

a = 6 t  ("s. 5ts 1  +he acceleration of a car (oving along 1x-axis is given by dislace(ent and velocity at  s is given by v = 9 ("s and x = 9 (. :alculate its osition and velocity at ti(e ! s. *ow (uch distance it covers in the ti(e interval t = ! s to t = 4 s. #cceleration of the article &that is (oving in a straight line ath' is given as 2 a =( 4 t + 2 ) m / s 0 ≤ t ≤ 2 , where $t% is ti(e in seconds. #t t = 0, v = 9("s and x = 4(. Find its velocity and dislace(ent as function of ti(e. 7hat is its velocity at t= sec8 5f the article is at # at t =  sec and at ; at t = ! sec, what is the length of #;8

12

A9erage 7elo"it8 an# a""eleration :onsider the case of <. 7hat is the average velocity and acceleration of the car in 13 ti(e interval t = ! s to t = 9 s8 14  +he x-coordinate of a article (oving along x-axis is given by  x =t 3− 3 t 2 . 7hat is the average velocity and acceleration in the ti(e interval &i' 0 to ! s and &ii' 0 to 3 s8 %5

Draw the grahs for the following functions

a e i

y = 4 x y =2 x −4 2

=−4 x + 6

ANSERS

! &  (

y =−6 x y =−4 x −6

" g ,

y = x + 4 2

=2 x =2 x − x −10 2

# '

y =−2 x + 4 2

= x + 2 x

1 2

t  + c

a

2

!

 y

3/ 2

+c

−1

"

2

#

+c

3

2

% a 3!9>

!

& 0.4

" !

1

e ln!.4

# 2

g >

' 6

−

i

( .

+ ln

a

4 3

! 1 4

' 0.!4

 ln

( )

"

33

3−5 ln

21

i -0.!4 -

/ ?

1 2

 4

 ln 3

0 

.

(

7 1 −ln

3

 ( 1

o 

#

4a

g 0.4

m 0.4

n

3

l 0.4

1

2 0.1

r

1

& -

3

e −1

,

4

 )

 10

10

e

s 0.!

t 0.4

2

 a. ! b. 0 c. 

2

3

 ρo a

2

3

3

π R  H 

4

2

2

π R  H  3

15

11

(+) 2

io

10

3

t 

t 

2

3

4 3

1%

1+

mgh

msT o

2

(

π ρ o R t  1−

2

2

1/

2

3

2

2 CR 3

)

72

3

v =6 t −0.6 t  + 2  s =3 t  −0.2 t  + 2 t  s= 3 t  −0.2 t  + 2 t +6  x =66 cmatt =5 s

> 1 (, 3 ("s, 6 (

12 35 3

13 ? ("s, ("s!

! 14 &i' ! av =−2 m / s i^ , ⃗aav = 0 2 > &ii' ! av = 0 , ⃗aav =−2 m / s i^ ⃗