Download OU (Osmania University) B.Sc Computer Science 5th Sem Integral Calculus Important Question Bank For 2021 Exam

I S

Subject Title: Integral Calculus

Prepared by: Ms Afreen Begum

Year: 3

Semester:

5

Updated on: 30.12.2020

Unit - I: AREAS AND VOLUMES

1.

Definition of Double integrals, Riemann sum, Integral over a Rectangle, Properties

of the integrals

2.

Integrate (i) 2 3( 2 + ) (ii) /2 1

3.

Compute

cos

, by changing the order of integration

4.

State Fubini's theorem.

5.

Change the order of integration and evaluate

+

6.

Let R=[-3,3]x[-2,2]. Without explicitly evaluate ( 5 + 2 )

7.

Find the area of the region ,using double integrals, bounded by y=2-

and x-

y=0,2x+y=0

8.

Integrate the function f(x, y)=3xy over the region bounded by

y=32x3 and y= .

9.

Evaluate ( 2 ) , where D is the region bounded by y=x2+2 and

y=2x2-2.

10. Use double integrals to find the area of the region bounded by the parabola y=2-

x2, and the lines x-y=0, 2x+y=0.

11. Find the volume of the region under the graph of f(x,y)=2-IxI-IyI and above the

xy-plane .

12.

1

Change of order of integration and evaluate (i) (2 - - )

0 0

2 4-2

2 4- 2

(ii)

(iii)

.

13. Find the volume of the region bounded by a graph of f

(x,y)=2 +

sin on top ,the xy-plane on the bottom and the planes

x=0,x=1,y=-1,y=2 on the sides.

14. Compute (5 | |)

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15. Discuss the types of elementary regions in the plane used in evaluating the double

integrals.

Unit-II TRIPLE INTEGRALS

16. Define Integral over a box, Riemann sum of triple integrals, triple integrals,

Fubini's theorem of triple integrals, Elementary region in space, types of triple

integrals.

17. Find the volume inside the capsule bounded by the paraboloids z=9-x2-y2 and

z=3x2+3y2-16.

18. Evaluate the triple integrals (i) [ 1,1] [0,2] [1,3] (i ) [ , ] [ , ] [ , ]

.

19. Evaluate 1 2 + .

0 1+

20. Integrate the given function over the indicated region W

(i) f(x,y,z)=2x-y+z, W is the region bounded by the cylinder z=y2, the xy-plane

and the planes x=0, x=1, y=-2, y=2.

(ii) f(x,y,z)= 8xyz; W is the region bounded by the cylinder y=x2, plane y+z=9

and the xy- plane.

21.

Change the order of integration of 1 1 2 ( , , )

to give the five

0 0 0

other equivalent iterated integrals.

22. Find the volume of the solid bounded by z=4-x2, x+y=2 and the coordinate planes.

23.

Find the volume of the solid bounded by the planes y=0, z=0,

2y+z=6 and the cylinder x2+y2=9.

24. Find the volume of the solid bounded by the paraboloid z=4x2+y2 and the cylinder

y2+z=2.

25. Find the volume of the solid over the function f(x,y,z)=4x+y and W is the region

bounded by x=

,y=z, x=y and z=0 .

26. Find the volume of the ellipsoid + + = 1.

27. Evaluate ( + 2 + )

.

28. Integrate the function f(x,y,z)=x+y over the region bounded by

+ 3 = 9 and y=0 , x+y=3.

29. Compute 3

where W is the regionin the first octant bounded by z=

+ ,x=0 ,y=0 and

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z=4.

30. Evaluate ( , , ) where B is the tetrahedron with the vertices (0,0,0),(1,0,0),(0,1,0) and

(0,0,1) and f(x,y,z)=1+xy.

Unit-III CHANGE OF VARIABLES

31. Change of variables and coordinate transformation, Jacobian of

double and triple integrals.

32. Evaluate ( 2 2)

, where D is the region in the

first quadrant bounded by the hyperbolas xy=1, xy=4 and

the lines y=x, y= x+2.

33.

Double integrals in polar coordinates, Cartesian coordinates, general coordinates

34. Change of variables in triple integrals, triple integrals in cylindrical coordinates,

spherical coordinates.

35.

Calculate the volume of the cone of height `h' and radius `a', in

which the cone is a solid W bounded by the surface

= 2 + 2

and the plane z=h. (using both cylindrical and spherical

coordinates).

36. If T(u,v)=(3u,-v),find the matrix A such that T(u,v)=A

.

37. Suppose T(u,v)=

,

,describe how T transforms the unit square [0,1]x[0,1]?

38. If T(u,v,w)=(3u-v,u-v+2w,5u+3v-w),describe how T transforms the unit cube [0,1]x[0,1]x[0,1]?

39. Evaluate the integral (2

)

by using substitutions u=2x-y , v=y.

40. Determine the value of ( + + ) where W denotes the solid region in the first octant

between the sphere

+ + =

+ + = where 0<a<b.

41.

2 3

If T(u,v)=

1 1

and is the parallelogram whose vertices are (0,0),(1,3),(-1,2) and (0,5)

. Determine D=T( ).

42. Find the area of the region inside both of the circle r=2a cos and r=2a sin where a is a

positive constant.

43. Find the volume of a ball of radius a using spherical coordinates.

44. Find the volume of a cone of radius a and height h using spherical coordinates.

45.

Evaluate

by using cylindrical coordinates.

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This post was last modified on 23 January 2021