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Area and Volume (Solved Questions)
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Page 1
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uestion ? ? sin(y
) y
x
(a)
os
( ) os ( ) os ( )
os
Jam MA 2017
Solution:
egion oun e y
x x an y x y
ow we evaluate the integral y hanging the or er of integral
? ? sin(y
) y
x
? siny
? x
y
? siny
,x-
y
? sin(y
) y y
, os y
-
os
uestion he value of the ou le integral ? ?
siny
y
y
x is
Jam MA 2016
Solution:
egion of integration is oun e y y y x an x x
hanging the or er of integration
? ?
siny
y
y
x ? ?
siny
y
x
y
?
siny
y
,x-
y
?
siny
y
, - y
? siny y
, os y-
uestion he value of the integral ?vx
y
x y
*(x y)
x x
y
x + is
(a) ( )
( )
( )
Jam MA 2013
uestion (a) valuate ? ? (e
)
y
v
x
Page 2
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uestion ? ? sin(y
) y
x
(a)
os
( ) os ( ) os ( )
os
Jam MA 2017
Solution:
egion oun e y
x x an y x y
ow we evaluate the integral y hanging the or er of integral
? ? sin(y
) y
x
? siny
? x
y
? siny
,x-
y
? sin(y
) y y
, os y
-
os
uestion he value of the ou le integral ? ?
siny
y
y
x is
Jam MA 2016
Solution:
egion of integration is oun e y y y x an x x
hanging the or er of integration
? ?
siny
y
y
x ? ?
siny
y
x
y
?
siny
y
,x-
y
?
siny
y
, - y
? siny y
, os y-
uestion he value of the integral ?vx
y
x y
*(x y)
x x
y
x + is
(a) ( )
( )
( )
Jam MA 2013
uestion (a) valuate ? ? (e
)
y
v
x
For more notes visit santoshifamily.com or call 8130648819
Solution
? ? (e
)
y
v
x
? ? e
y
v
x
? *
e
+
v
x
? (
e
e
v
) x
?
e
x
? e
v
x
e
? e
( z ) z
,put x z
-
e
? ze
z
e
*
ze
?
+
*
+
e
*
e
e
+
e
e
e
e
( e
)
uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y
) x
Jam MA 2011
Solution:
omain of integration is oun e y the following urves
y x y x
x x
point of interse tion of the urve y x
an y x we get when
x
x
x
x
(x )(x )
x
Page 3
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uestion ? ? sin(y
) y
x
(a)
os
( ) os ( ) os ( )
os
Jam MA 2017
Solution:
egion oun e y
x x an y x y
ow we evaluate the integral y hanging the or er of integral
? ? sin(y
) y
x
? siny
? x
y
? siny
,x-
y
? sin(y
) y y
, os y
-
os
uestion he value of the ou le integral ? ?
siny
y
y
x is
Jam MA 2016
Solution:
egion of integration is oun e y y y x an x x
hanging the or er of integration
? ?
siny
y
y
x ? ?
siny
y
x
y
?
siny
y
,x-
y
?
siny
y
, - y
? siny y
, os y-
uestion he value of the integral ?vx
y
x y
*(x y)
x x
y
x + is
(a) ( )
( )
( )
Jam MA 2013
uestion (a) valuate ? ? (e
)
y
v
x
For more notes visit santoshifamily.com or call 8130648819
Solution
? ? (e
)
y
v
x
? ? e
y
v
x
? *
e
+
v
x
? (
e
e
v
) x
?
e
x
? e
v
x
e
? e
( z ) z
,put x z
-
e
? ze
z
e
*
ze
?
+
*
+
e
*
e
e
+
e
e
e
e
( e
)
uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y
) x
Jam MA 2011
Solution:
omain of integration is oun e y the following urves
y x y x
x x
point of interse tion of the urve y x
an y x we get when
x
x
x
x
(x )(x )
x
Free coachi
For more notes visit santoshifamily.com or call 8130648819
So ( ) an ( ) are the points of interse tion
ow the given integral mo ify y hanging the or er of integration we get
? ? f(x y) x
v
y
? ? f(x y) x
v
y
Something skip follow book
uestion hanging the or er of integration of ? ? f(x y) y
x
(a) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
Jam CA 2011
Solution:
? ? f(x y) y
x
he region of integration is oun e y y y x x x
after hanging the or er the integration hanges to
? ? f(x y) x
y
? ? f(x y) x
y
uestion et e the trianle oun e y the y axis the line y hen the value of the integral
?
os y
y
x y is
(a)
( ) ( )
( )
Jam MS 2011
Solution
? ?
os y
y
x
y
?
os y
y
,x-
y
? os y y
,siny-
uestion ( ) hange the or er of integration in the integral ? ? f(x y) y
v
x
Jam MA 2010
Solution:
he omain of the ou le integration is oun e y the urves y x y v x x x
Page 4
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uestion ? ? sin(y
) y
x
(a)
os
( ) os ( ) os ( )
os
Jam MA 2017
Solution:
egion oun e y
x x an y x y
ow we evaluate the integral y hanging the or er of integral
? ? sin(y
) y
x
? siny
? x
y
? siny
,x-
y
? sin(y
) y y
, os y
-
os
uestion he value of the ou le integral ? ?
siny
y
y
x is
Jam MA 2016
Solution:
egion of integration is oun e y y y x an x x
hanging the or er of integration
? ?
siny
y
y
x ? ?
siny
y
x
y
?
siny
y
,x-
y
?
siny
y
, - y
? siny y
, os y-
uestion he value of the integral ?vx
y
x y
*(x y)
x x
y
x + is
(a) ( )
( )
( )
Jam MA 2013
uestion (a) valuate ? ? (e
)
y
v
x
For more notes visit santoshifamily.com or call 8130648819
Solution
? ? (e
)
y
v
x
? ? e
y
v
x
? *
e
+
v
x
? (
e
e
v
) x
?
e
x
? e
v
x
e
? e
( z ) z
,put x z
-
e
? ze
z
e
*
ze
?
+
*
+
e
*
e
e
+
e
e
e
e
( e
)
uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y
) x
Jam MA 2011
Solution:
omain of integration is oun e y the following urves
y x y x
x x
point of interse tion of the urve y x
an y x we get when
x
x
x
x
(x )(x )
x
Free coachi
For more notes visit santoshifamily.com or call 8130648819
So ( ) an ( ) are the points of interse tion
ow the given integral mo ify y hanging the or er of integration we get
? ? f(x y) x
v
y
? ? f(x y) x
v
y
Something skip follow book
uestion hanging the or er of integration of ? ? f(x y) y
x
(a) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
Jam CA 2011
Solution:
? ? f(x y) y
x
he region of integration is oun e y y y x x x
after hanging the or er the integration hanges to
? ? f(x y) x
y
? ? f(x y) x
y
uestion et e the trianle oun e y the y axis the line y hen the value of the integral
?
os y
y
x y is
(a)
( ) ( )
( )
Jam MS 2011
Solution
? ?
os y
y
x
y
?
os y
y
,x-
y
? os y y
,siny-
uestion ( ) hange the or er of integration in the integral ? ? f(x y) y
v
x
Jam MA 2010
Solution:
he omain of the ou le integration is oun e y the urves y x y v x x x
Free AM
? ? f(x y) y
v
x
? ? f(x y) x
y
? ? f(x y) x
v
y
uestion et ? ? xy
v
v
x
y ? ? xy x
v
y
hen using the transformation x r os
y rsin integral is equal to
(a) ? ? r
sin r
( ) ? ? r
sin r
( ) ? ? r
sin r
( ) ? ? r
sin r
Jam CA 2010
Solution
egion of integration is oun e y urves
x v y
x v y
y y
for first integral
uestion valuate integral ? ?
siny
y
y
x
is
( ) ( )
( ) ( )
Jam CA 2010
Solution
egion of integration is oun e y urves
y x y
x x
hanging the or er
? ?
siny
y
y
x
? ?
siny
y
x
y
? ?
siny
y
x
y
? siny y
uestion he value of
? ? e
(
)
x
y
equals
(a)
( )
( )
( )
Jam MS 2010
Solution:
Page 5
For more notes visit santoshifamily.com or call 8130648819
uestion ? ? sin(y
) y
x
(a)
os
( ) os ( ) os ( )
os
Jam MA 2017
Solution:
egion oun e y
x x an y x y
ow we evaluate the integral y hanging the or er of integral
? ? sin(y
) y
x
? siny
? x
y
? siny
,x-
y
? sin(y
) y y
, os y
-
os
uestion he value of the ou le integral ? ?
siny
y
y
x is
Jam MA 2016
Solution:
egion of integration is oun e y y y x an x x
hanging the or er of integration
? ?
siny
y
y
x ? ?
siny
y
x
y
?
siny
y
,x-
y
?
siny
y
, - y
? siny y
, os y-
uestion he value of the integral ?vx
y
x y
*(x y)
x x
y
x + is
(a) ( )
( )
( )
Jam MA 2013
uestion (a) valuate ? ? (e
)
y
v
x
For more notes visit santoshifamily.com or call 8130648819
Solution
? ? (e
)
y
v
x
? ? e
y
v
x
? *
e
+
v
x
? (
e
e
v
) x
?
e
x
? e
v
x
e
? e
( z ) z
,put x z
-
e
? ze
z
e
*
ze
?
+
*
+
e
*
e
e
+
e
e
e
e
( e
)
uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y
) x
Jam MA 2011
Solution:
omain of integration is oun e y the following urves
y x y x
x x
point of interse tion of the urve y x
an y x we get when
x
x
x
x
(x )(x )
x
Free coachi
For more notes visit santoshifamily.com or call 8130648819
So ( ) an ( ) are the points of interse tion
ow the given integral mo ify y hanging the or er of integration we get
? ? f(x y) x
v
y
? ? f(x y) x
v
y
Something skip follow book
uestion hanging the or er of integration of ? ? f(x y) y
x
(a) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
( ) ? ? f(x y) x
y
? ? f(x y) x
y
Jam CA 2011
Solution:
? ? f(x y) y
x
he region of integration is oun e y y y x x x
after hanging the or er the integration hanges to
? ? f(x y) x
y
? ? f(x y) x
y
uestion et e the trianle oun e y the y axis the line y hen the value of the integral
?
os y
y
x y is
(a)
( ) ( )
( )
Jam MS 2011
Solution
? ?
os y
y
x
y
?
os y
y
,x-
y
? os y y
,siny-
uestion ( ) hange the or er of integration in the integral ? ? f(x y) y
v
x
Jam MA 2010
Solution:
he omain of the ou le integration is oun e y the urves y x y v x x x
Free AM
? ? f(x y) y
v
x
? ? f(x y) x
y
? ? f(x y) x
v
y
uestion et ? ? xy
v
v
x
y ? ? xy x
v
y
hen using the transformation x r os
y rsin integral is equal to
(a) ? ? r
sin r
( ) ? ? r
sin r
( ) ? ? r
sin r
( ) ? ? r
sin r
Jam CA 2010
Solution
egion of integration is oun e y urves
x v y
x v y
y y
for first integral
uestion valuate integral ? ?
siny
y
y
x
is
( ) ( )
( ) ( )
Jam CA 2010
Solution
egion of integration is oun e y urves
y x y
x x
hanging the or er
? ?
siny
y
y
x
? ?
siny
y
x
y
? ?
siny
y
x
y
? siny y
uestion he value of
? ? e
(
)
x
y
equals
(a)
( )
( )
( )
Jam MS 2010
Solution:
Free M
? ? e
r
r
? r e
r
* e
+
uestion (a) in the area of the smaller of the two regions en lose etween
x
y
an y
x
( ) valuate ? ? e
x
y
Jam MS 2010
Solution
rea ? ? x
(
)
y
v
v
? ,x-
v
y
v
v
? (
v
v y
y
) y
v
v [? v y
y
v
y
]
v
v [
y
v y
sin
y
v
]
v
v
v v sin
v
v
v
v
Solution ( )
he region of integration is oun e y x x y y y an shown in figure
hanging the or er
? ? e
.
/
y
x
? 0 e
e
(x )1
x
? e
(x )e
x
e
? (xe
e
)
x
e
Read MoreFAQs on Area and Volume (Solved Questions)
| 1. What is the formula for finding the area of a rectangle? |  |
Ans. The formula for finding the area of a rectangle is length multiplied by width.
| 2. How do you calculate the volume of a cylinder? | |
Ans. The volume of a cylinder can be calculated by multiplying the area of the base (circle) by the height of the cylinder.
| 3. What is the formula for finding the volume of a rectangular prism? | |
Ans. The formula for finding the volume of a rectangular prism is length multiplied by width multiplied by height.
| 4. How can I find the area of a triangle? | |
Ans. To find the area of a triangle, you can use the formula: 1/2 multiplied by base multiplied by height.
| 5. How do you calculate the volume of a cone? | |
Ans. The volume of a cone can be calculated by multiplying the area of the base (circle) by the height of the cone and then dividing the result by 3.
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