Page 42 - Engineering Mathematics Workbook_Final

Calculus

 1   u v 2u v  
−
+
)
−
(d) (cos1 cos4 [JAM CA 2007] (d) 0  4 − u u  /2  f ,    dv du
2 3  3 3 
234. Consider the double integral 236. The area of the region bounded by the
0  1 x  2 x f ( ,x y dy dx . After curves x = 2y and y = 2x is
)
+
2
2
reversing the order of the integration, the
integral becomes (a) 1 (b) 2
3 3
(a)
0   0 y− 2 f ( , x y dx dy + 1   0 1 f ( , x y dx dy (c) 4 (d) 4
)
)
2
1
)
+ 2  3 y  1 f ( , x y dx dy 3
(b) [JAM CA 2008]
)
0   0 y f ( , x y dx dy + 1   0 1 f ( , x y dx dy237. The value of the integral
)
2
1
)
+ 2  3 y−  1 2 f ( , x y dx dy 0  3 0  3x dy dx is
x + y 2
2
(c)
0   0 y f ( , x y dx dy + 1   0 y f ( , x y dx dy (a) 3log 2 + 3
)
(
)
)
2
1
)
+ 2  3 y−  1 2 f ( , x y dx dy
(
)
(b) 3log 2 − 3
(d)
0   0 y− 2 f ( , x y dx dy + 1   0 y f ( , x y dx dy
)
)
2
1
)
+ 2  3 y  1 f ( , x y dx dy (c) 3log 2
)
(
3
(d) log 2 + 3 [JAM CA 2008]
[JAM CA 2008] 2
235. The double integral 238. Changing the order of integration of
0  2 x  4 x f ( , x y dydx under the −  1 1  1 x 2 f ( , x y dy dx gives
)
−
)
−
−
transformation u = x y , v = y − 2x − 1 x 2
+
)
+
1 y
is transformed into (a) 0  1  1 y f ( ,x y dy dx +
−
 u v 2u v 
−
+
(a) 0  4 u  u /2  f  ,     dv du −  0 1  1 y 2 f )
−
 3 3  − 1 y 2 ( , x y dy dx
−
+
−
)
+
(b) 3 0  4 u  u /2  f   u v , 2u v      dv du (b) 0  1  1 y f ( ,x y dy dx
−
 3 3  1 y
)
+
−
−
1 4 u   u v 2u v   − −  0 1  1 y 2 f ( ,x y dy dx
(c) 0  u  /2  f ,    dv du − 1 y 2
−
3  3 3 

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