R.M.K . COLLEGE OF ENGINEERING AND TECHNOLOGY

R.S.M NAGAR, PUDUVOYAL-601206

MA2111 – MATHEMATICS-I

ASSIGNMENT QUESTIONS (REGULATION 2008)

UNIT –V

MULTIPLE INTEGRAL

DOUBLE INTEGRALS IN CARTESIAN CO-ORDINATES:

PART-A1.Evaluate the following integrals:

(i) ∫34∫1

2( x+ y )−2 dxdy (Au, May 2006)

(ii) ∫2a

∫2b dxdyxy (Au, Nov 2004,2008)

(iii) ∫24

∫12 dxdy

x2+ y2

(iv) ∫01∫1

2x ( x+ y )dydx

(v) ∫01

∫01 dxdy

√(1−x2 )(1− y 2)

(vi) ∫0

∞

∫0

∞e−( x2+ y2)dxdy (Au, May 2008)

(vii) ∫0

5

∫0x2

x ( x2+ y2 )dxdy

(viii) ∫01∫x

√xxy ( x+ y )dydx (Au, May 2009)

(ix) ∫0

1

∫0√1+x2 dydx

1+x2+ y2

(x) ∫0

a

∫0

√a2− y2 √a2−x2− y2dxdy

2. Sketch roughly the region of integration of ∫0b∫0

ab(b− y )

f (x , y )dxdy .

(Au,May 2008)

3. Sketch roughly the region of integration for the double integral ∫01∫0

xf (x , y )dydx .

(Au,Dec 2007)

4. Shade the region of integration ∫0

a

∫√ax−x2√a2− x2 dydx .

5.Find the limits of integration in the double integral ∬R

f (x , y )dxdy,where R is in the

first quadrant and bounded by x=1,y=0,y2=4x. (Au, Dec 2007)

PART-B

6. Evaluate ∬(1−xy )dxdy in the region bounded by the line y=x-1 and the parabola y2=2x-6. (Au,April 2004)

7. Evaluate ∬ ydxdy over the part of the plane bounded by the line y=x and the parabola y=4x-x2.

8. Evaluate ∬R

x2dxdy ,where R is the region in the first quadrant bounded by the lines

x=y, y=o, x=8 and the curve xy=16.

9. Evaluate ∬R

xydxdy ,where A is the domain bounded by x-axis, ordinate x=2a and

the curve x2=4ay.

10.Find the value of ∬ xydxdy taken over the positive quadrant of the

ellipse x2

a2+ y

2

b2=1

. (Au,Nov 2008)

11.Evaluate ∬( x+ y )2 dxdy over the area bounded by the ellipse x2

a2+ y

2

b2=1

.

12.Evaluate ∬R

ydxdy where ,R is the region bounded by the parabolas y2=4x and

x2=4y.

13. Evaluate ∬ xydxdy over the positive quadrant of the circle x2+y2=1.

DOUBLE INDTEGRALS IN POLAR CO-ORDINATES:

PART-A

14. Evaluate the following integrals:

(I) ∫0π2∫0

2rdrd θ (Au,May 2005)

(ii) ∫0π2∫0

sin θrdrd θ

(iii) ∫0π∫0

cosθ3 r2drd θ

(iv) ∫0π2∫0

acos θr √a2−r2drd θ

(v) ∫0π∫0

a(1−cos θ)r2sin θdθdr

(vi) ∫−π

2

π2 ∫0

2cosθr 2drd θ

PART-B

15. Evaluate ∬ r2drd θ over the area bounded between the circles r=2 cosθ and r=4cosθ . (Au, May 2008)

16. (i) Evaluate ∬ r 3 drd θ over the area bounded between the circles r=2cosθ and r=4cosθ. (Au, Dec 2007)

(ii) Evaluate ∬ r3drd θ over the area bounded between the circles r=2sin θand r=4sin θ. (Au, May 2005)

17. Evaluate ∬ r sin θdrd θ over the cardioid r=a(1-cosθ ¿ above the initial line.

18. Evaluate ∬ rdrd θ

√a2+r2 over one loop of the lemniscate r2=a2cos2θ

.

CHANGE OF ORDER OF INTEGRATION

PART-A

19. Change the order of integration in the following integrals:

(i) ∫0a∫0

xdydx

(ii) ∫0a

∫y

a x+ yx2+ y2

dxdy

(iii)∫02∫0

xf (x , y )dydx

(iv)∫0

1

∫x2

2− xf ( x , y )dydx

PART-B

20. Change the order of integration and then evaluate the following integrals:

(i) ∫0

a

∫a− y√a2− y2 ydxdy (Au, May 2009)

(ii) ∫0

∞

∫0

xxe

−x 2

ydydx (Au, May 2008)

(iii) ∫04

∫y

4 x

x2+ y2dxdy

(Au,Dec 2008)

(iv) ∫0

3

∫1

√4− yx+ ydxdy

(v) ∫0

1

∫y2

2− yxydydx

(Au, May 2008)

(vi) ∫0

a

∫x2

a

2a− xxydxdy

(Au, Nov 2004, Dec 2007, May 2007)

(vii) ∫0a∫x

a( x2+ y2)dydx (Au, Dec 2006)

(viii) ∫0

1

∫y

√ yx

x2+ y2dxdy

(Au, May 2005, 2006)

(ix) ∫0

1

∫x√2− x2 x

√x2+ y2dxdy

(Au,May 2007)

(x) ∫0

4 a

∫x2

4 a

2√axdydx

(Au,Nov 2008)

(XI) ∫0a

∫y

a xdxdy

( x2+ y2 )

(xii) ∫03∫0

√4− y(x+ y )dxdy

CHANGE OF VARIABLES BETWEEN CARTESIAN AND POLAR CO-ORDINATES

PART-A

21. Change into polar coordinates, the integral ∫−a

a

∫−√a2−x 2√a2−x2 dydx .

.

PART-B

22. Express ∫0a∫y

a x2 dxdy( x2+ y2 ) in polar coordinates and then evaluate it.

(Au,Dec 2007,May 2009)

23. Change into polar coordinates and evalulate ∫0

∞

∫0

∞e−( x2+ y2)dydx . Hence show

that ∫0

∞e−x

2

dx=√ π2.

24. Evaluate ∬√ 1−x2− y21+x2+ y2

dxdy over the positive quadrant of the circle x2+y2=1.

25. Evaluate ∫0

2

∫0√2 x− x2 xdydx

√x2+ y2by changing to polar coordinates.

TRIPLE INTEGRALS IN CARTESIAN CO-ORDINATES

PART-A

26. Evaluate the following integrals:

(i) ∫0

1

∫0

√1−x2∫0√1−x 2− y 2

xyzdxdydz

(ii) ∫−1

1 ∫0

z∫x−z

x+ z( x+ y+z )dxdydz

(iii) ∫0log 2∫0

x∫0x+ yex+ y+ zdxdydz (Au,May 2008)

(iv) ∫01∫0

2∫03xyzdxdydz (Au, May 2008, May 2009)

AREA AS DOUBLE INTEGRAL

PART-B

27. Find the area of the ellipse x2

a2+ y

2

b2=1

by double integration.

28. Find the smaller of the areas bounded by y=2-x and x2+y2=4. (Au,Nov 2008)

29. Find using double integration the area of the cardiod r=a(1+cosθ).

30. Find the area of the region D bounded by the parabola y=x2 and x=y2.

(Au, May 2007)

31. Show that the area between the parabolas y2=4ax and x2=4ay is 16a2/3.

32. Find the area lying inside the cardioid r=a(1+cosθ) and outside the circle r=a.

33. Find by double integration the area lying inside the circle r=asin θ and

outside the cardioid r=a(1-cosθ).

34. Calculate the area included between the curve r=a(secθ +cosθ) and

its asymptote.

35. Find the area lying between the parabola y=4x-x2 and the line y=x.

36. Find the smaller of the areas bounded by the ellipse 4x2+9y2=36 and the straight line 2x+3y=6.

VOLUME AS TRIPLE INTEGRAL

PART-B

37. Find by triple integration, the volume in the positive octant bounded by the coordinates planes and the plane x+2y+3z=4.

38. Evaluate ∭dxdydz where ,V is the region of space inside the cylinder x2+y2=4, that is bounded by the planes z=0 and z=3. (Au, Dec 2008)

39. Find by triple integral the volume of the tetrahedron bounded by the planes

x=0,y=0,z=0 and xa+ yb+ zc=1

. (Au, May 2005,2007,Dec 2006)

40. Evaluate ∭dxdydz , where Vis the finite region of space (tetrahedron) formed by the planes x=0,y=0,z=0 and 2x+3y+4z=12.

41. Express the volume of the sphere x2 +y2 +z2 =a2 as a volume integral and hence evaluate it. (Au, Dec 2007)

42. Find the volume of that portion of the ellipsoid x 2

a2+ y

2

b2+ z

2

c2=1

which lies

in the first octant using triple integration. (Au, May 2007)

43. Find the volume bounded by the cylinder x2+y2=4 and the planes y+z=4

and z=0.

44. Evaluate ∭dxdydzover the volume cut off the sphere x2+y2+z2=a2 by

the cone x2+y2=z2.

45. Evaluate ∭ xyz dxdydz over the positive octant of the sphere

x2+y2+z2 = a2 .

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