Page 53 - Engineering Mathematics Workbook_Final

Calculus

1
320. The integral   (x + y + 10 )dxdy
2 D

where D denotes the disc: x + 2 y  2 4

, evaluates to __________

[GATE-16-EC-SET 1]
[GATE-20-EC]
321. The region specified by 325. The area of an ellipse represented by
    2 2

 
z
 ( , ,Z ) :3   5,    , 3   4.5  x y
 8 4  an equation a 2 + b 2 = 1 is
in cylindrical coordinates has volume  ab

of __________ (a) ab (b)
4

[GATE-16-EC-SET 1] (c)  ab (d) 4 ab
2 3
322. A triangle in the xy-plane is bounded [GATE-20-CE-SET1]
by the straight lines 2x = 3 , y y = 0 326. The volume of
and x = 3. The volume above the  1 dx dydz where R is

triangle and under the plane ( x + 2 y + 2 z 2 ) 3/ 2
+ +
x y z = 6 is __________ the region bounded by
2
2
2
2
x + y + z = b
[GATE-16-EC-SET 3]
)

( /
ln
(a) ( / b a (b) 4 ln b a )
)
)


( /
( /
323. A three dimensional region R of (c) 2 ln b a (d) 8 ln b a
finite volume is described by 327. The volume of
 ( x + 2 y + 2 z 2 ) dx dydz taken over the
2
2
z
x + y  z 3 ;0   .
1
2
volume enclosed by x + y + z = 1 is _
2
2
Where , , y z are real. The volume of R 4 8
x
(upto two decimal places) is ______ (a) 5 (b) 5
2 
[GATE-17-EC; SESSION 1] (c) (d)
5 5
324. For the solid S shown below, the 328. The volume of the solid surrounded
value of  xdxdydz (rounded off to by the surface 2/3
2/3
2/3
s  x  +  y  +  z  = 1
 
 
 
two decimal places) is  a   b   c 

4 abc abc
(a) (b)
35 35

2 abc  abc
(c) (d)
35 35
[EE, ESE-2019]

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