University of Delhi 2009-2nd Year B.Sc Computer Science (Hons)/ III sem /s / NS -203 calculus-II - Question Paper
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1977B.Sc. (Hons.) II SemJNS G
COMPUTER SCIENCE Paper 203Calculus II (New Course)
(Admissions of 2001 and onwards)
Time : 3 Hours Maximum Marks : 75
(Write your Roll No. on the top immediately on receipt of this question paper.)
All questions are compulsory.
1. (a) Use the max-min inequality to show that if f is integrable then :
b
f(x) > 0 on [a, 6] J f(x) dx>0
b
fix) < 0 on {a, b\ => f f(x}dx<0.
(&K Show that if f is continuous on [a, b] a * b and
if
O
J f(x)dx = 0
then f{x) = 0 at least once in [a, 6]. 5
The velocity of a particle moving in space is
= (t3 + At) i + tj + 2t2k
dt
Find the particles position as a fn. of i if
- I + j when t = 0. 5
(b) The region bounded by the curve y = *fx, the x-axis and the line x = 4 is revolved about the jc-axis to generate a solid. Using Shells formula find the volume of the solid. 5
Find the volume of a solid using slicing method when the solid lies between planes perpendicular to x-axis at x ~ - 1 and x'= 1. The cross-sections perpendicular to the axis between these planes are
vertical squares whose base edges run from the semicircle y = -~ - jc2 to the semicircle y = {l-x2- 5
(}}) . Find the area of the surface generated by revolving the curve y = x*3, 0<x<~7 about the x-axis.
4. ($' Find the centre of mass of a thin plate of
/
/
density 5-3 bounded by the lines x - 0, y x and the parabola y 2 ar in the first quadrant. 5
(6) Evaluate :
JJ e*2 * -y2 dydx
H
where R is the semicircular region bounded by the x-axis and the curve y ~ I ~ x2 5
5. (py Liet D be the region in xyz-space defined by the
inequalities :
1 < x < 2, 0 < xy < 2, 0 < z < 1.
D
by applying the transformations :
u - x, v = xy and w = 3z
and integrating over the appropriate region G in UVW plane. 5
y
Find an analytic function whose real part is given function U(x, y) ~ x - xy. 5
() Find the images of x = constant and y - constant under f(z) ~ sin z. 5
() State Cauchy integral theorem. Use it to find the
f * +
4
if J , n
value of J pf + 2Z15 dz, if c is the circle i2+l|=l. 5
(c) Use Residue theorem to evaluate :
sin nx cos nx
2 n
converges to the periodic fn. f in ]- n, 7t[
where
x2 + x for - 7t < x < n n2 for x - n
/<*) =
(b) Expand in a series of sines and consines of multiple angles of x, the periodic fn. f with period 2n defined
-1 for - < x < 0 .
/(*) =
1 for 0 < x < 71
Also calculate the sum of the series at ft
1977 5 600
2 '
x -
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