Page 48 - Engineering Mathematics Workbook_Final

Calculus

8 288. A parametric curve defined by
(d) ( 10 3/ 2 + ) 2
27   u    u 
x cos   , y = sin   in the range
284. Let f be increasing, differentiate  2   2 


function. If the curve y = f ( ) x passes 0 u  1 is rotated about the X-axis by
0
through (1, 1) and has length 360 . Area of the surface generated is

2 1 (a) (b) 
1 x 
L =  1+ 2 dx  2, then curve 2
1 4x (c) 2 (d) 4
is __________ [GATE-17-ME]

)
x −
(a) y = ln ( ) 1 289. Let W = f ( ,x y , where x and y are
functions of t. Then, according to the
−
(b) y = 1 ln x dw
)
(c) y = ln 1+ ( x chain rule, dt is equal to

+
x
(d) y = 1 ln ( ) (a) dw dx + dw dt
dx dt dy dt
285. The length of the arc  w x   w x 
)
x = ( a t − sint ), y = a (1 cost between (b)  x t  +  y t 
−
t = 0 to t = 2 is _____ (c)  w dx +  w dx

(a) 8a (b) 4a  x dt  y dt
dw x  dw x 
(c) 4 2a (d) 2 2a (d) +
dx t  dy t 
)
+
286. The length of the arc r = a (1 cos
290. The surface area obtained by
between = 0 to  is _____ revolving y = 2x for x 0,2 about y –

287. Consider a spatial curve in three-
axis is ____
dimensional space given in parametric
(a) 2 5 (b) 4 5
from by x ( ) t = cos , y ( ) sin ,t = t
t
(c) 2 5 (d) 4 5
2 

z ( ) t = t , 0 t 
 2 291. The surface area generated by
The length of the curve is ___________ rotations

3
3

[ME, GATE-2015 : 2 MARKS] x = a cos  , y = a sin  ,0   
about y- axis

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