North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH-211 (Calculus of Several Variables) - Question Paper

5x2 + 6xy + 5y2 = 8
29) If a,b,c,are +ve numbers obtain the extreme value of f(x,y,z) = xa yb zc subject to the condition
x+y+z =1
30) obtain extreme values of f(x,y) = xy ( a-x-y)
Unit IV ( Double and Triple Integrals)
I) Objective ques. ( two marks each)
1) describe double integral ??
R
f ( x, y )dR
2) Evaluate : (x y)dydx
x
? ? +
2
1 0
2
11
3) Evaluate : ? ?(x + y )dxdy
2
1
1
0
2 2
4) Evaluate : e dxdy
x
x
y
? ?
1
0 0
2
5) Evaluate : ? ? +
2
1 0
2 2
x
x y
dxdy
6) Evaluate : : (x y )dxdy
a b
? ? +
0 0
2 2
7)Evaluate : xydxdy
y
? ?
0
4
0
8) Evaluate : ?? xydxdy over the rectangle bounded by x=2, x=5, y=1, y=2.
9) Evaluate : ? ? +
1
0
1
0
dx ( x y )dy
10)Evaluate : : ? ?
1 -
0
1
0
x
xydydx
11) describe triple integral ???
v
f ( x, y,z )dv
12) Evalute : ; ? ? ? + +
1
0
1
0
1
0
( x y z )dxdydz
13) Evaluate ; ? ? ?
1
0
2
0
2
1
xyzdxdydz
14) Evaluate : ( ) dx
x y
dy ? ? ??
?
??
?
+
4
3
2
1
2
15)Evaluate : ? ?(x - y)dxdy
2
1
2
1
II ) Multiple option ques. (1 marks each)
select the accurate choice from the provided choices.
1) The area of region bounded by the circle x two + y2 = a2 is
a) ? 2a unit b) ?a2 unit 2
c) ?a3 unit three d) ?a unit
2) The area of the ellipse 1
16 9
2 2
x + x = is
a) 12 ? unit two b) 12 unit
c) 12 ? unit three d) 12 ?
3) The quantity of the sphere x2 + y2 + z two = a2 is
a) ?a2 unit two b) ?a3 unit
c) 3
3
4 ?a unit three d) ?a unit
12
4) The quantity of ellipsoid 1
1 four 9
2 two 2
x + y + z = is
a) ?
3
4 unit b) 2
3
4 ? unit 2
c) ?
3
24 unit three d) ?
3
12 unit 3
III) Theory and Examples ( six marks each)
1) a) Draw a sketch of the region of integration
? ?
-
2 +
1
2
2
( , )
x
x
f x y dxdy
b) Evaluate : ? ? ?
= = =
+ +
3
0
2
0
1
y x z 0
( x y z )dzdxdy
2) a) Draw a sketch of the region of integration ,? ?
4 =
0
25
0
2
dx f ( x, y )dy
x
c) Evaluate : ? ? ?
= = =
1
0
1
0
2
1
2
x y z
x yzdzdydx
3) a) Draw a sketch of the region of integration ? ?
2 -
0
4 y
y
dy f ( x, y )dx
b) Evaluate : ? ? ?
1 - - -
0
1
0
1
0
x x y
xyzdxdydz
4) a) Draw a sketch of the region of integration ? ?
2
1
2
2
x
x
dx f ( x, y )dy
b) Evaluate ???( x + y + z )dxdydz over the tetrahedron x=0 , y =0, z =0 and x+y+z=1
5) a) change the order of integration and hence evaluate
? ? +
a a
y x y
xdxdy
0
2 2
b) Evaluate : ? ? ?
- - -
- - -
a x x y
x y z
dxdydz
0
1
0
1
2 two 2
2 two 2
1
6) a) Evaluate : ?? xydxdy over the region in the positive quadrant for which x+y = 1
b) Evaluate : ? ? ?
+
+ +
a x x y
ex y zdxdydz
0 0 0
7)a) Using double integral obtain the area of the circle x2 + y2 = a2
b) Evaluate : ? ? ?
1 - +
0
1
0 0
x x y
ezdxdydz
8) a) change the order of integration ? ?
8 eight -
0 x
y
dxdy
y
e
13
b) Evaluate ???( x + y + z +1)3
dxdydz over the region x = 0, y = 0,z = 0,x + y + z = 1
9) a) change the order of integration ? ?
2
1
2x
x
f ( x, y )dxdy
b)Evaluate ; ? ? ?
aa-xa-x- y
x dxdydz
0 0 0
2
10)a) obtain the area of ellipse one 2
2
2
2
+ =
b
y
a
x , using double integration.
b) Evaluate : ? ? ?
1 - +
0
1
0 0
x ( x y )2
xdxdydz
11) a) Using double integration obtain the area of the region bounded by the parabola’s y2 = 4x and
x2 = 4y
b) obtain the quantity of the sphere of radius 5.
12) a) Evaluate ?? e2x+3ydxdy over the triangle bounded by x =0, y = 0, and x+y = 1
b) obtain the quantity of the tetrahedron bounded by the co-ordinate planes and the plane
+ + = 1
c
z
b
y
a
x
13)a) Evaluate ?? ydxdy over the area bounded by y =x two and x+y=2
b) obtain the quantity of the region bounded by x =0 ,y = 0, z = 0 and x+y+z=1
14) a) Evaluate : ?? x y dxdy two 2 over the region x2 + y2 = 1
b) Using triple integration obtain the quantity of the sphere of radius a.
15) a) Evaluate ? ? ( )
-
+ - -
1
0
1
0
2 2
2
1 1
x
e y x y
dxdy
b) obtain the quantity bounded by the cylinder x2 + y2 = four and the planes y+z =3, z = 0
III) Compulsory Examples ( two marks each)
1) obtain the area of square bounded by lines x =0, x =1, y = 0, y = one using double integration.
2) Evaluate ? ? ?
= = =
2
x0 0 0
x
y
y
z
e pdxdydz ,where p is a real.
3) obtain the area bounded by the lines x=0 ,x=a,y =0, y = b.
4) obtain the quantity of region bounded by the planes
,x=0, x = 1, y = 0 , y = 1, z = 0 ,z =1.
5) obtain the area of the circle of radius one using double integration.
6) Evaluate ? ?
1
2
2
0
x3 y3dxdy
7) Evaluate : ? ? +
1
0
1
0
( x y )dxdy
8) Evaluate: ? ?
b
a
b
a
xydxdy

Earning:   Approval pending.