CHAPTER 5 : MULTIPLE INTEGRALS

CHAPTER 5 : MULTIPLE INTEGRALS

1. If R



0,6

 

 0,4



, use a Riemann sum with ^{m3, n2}to estimate the value of ^xydA



R . Take the sample points to be the (a) lower left corners,

(b) upper right corners of the rectangles.

2. Evaluate the double integral by first identifying it as the volume of a solid.

(a) 5 xdA,



R ^ R

 

x,y



|0x5 ,0 y3



. (b) 4 2ydA,



R ^{ R}



^0,1

 

^{ 0,1}



. 3. Calculate the iterated integral.

(a) ⁴

 

^.

2 1

1

2

2 y dydx

 

x



 (f) ^.

2 ln

0 5 ln

0

2 dxdy

e ^{x y}

 

^

(b) ² _{sin .}

0 2 0

dydx y

 

x



(g) ^.

1

1

0 1

0 2 2 dydx

y x

 

_xy _

(c) 2  .

2

0 1

0

8 dxdy y

 

x^{ (h) ln2 .}

0 1 0

2 dydx xye^{y x}

 

(d) ^.

1

0 2

1

y dydx xe^x

 

⁽ⁱ⁾

 ^{sin sin}  ^.

2 0

3 0

dydx x y-y

 

x

 

(e)  ¹  ^.

2

1 1

0

2 dxdy y

 

x_

4. Calculate the double integral.

(a) cosx 2ydA,



R ^

 







    

 x,y |0 x  ,0 y 2

R .

(b) 2 ₁ ^,

2

x dA xy



R _ ^R

 

^x,y



^{|0x1 ,3y3}



^.

(c) ₁¹ 2 ^,

2

y dA x



R _^{ R}

 

^x,y



^{|0x1 ,0y1}



^.

(d) xsinx ydA,



R ^{ }_^ _^_^ _^

,3 6 0

,

0  

R .

(e) ₁ ^x_xy ^dA,



R _ ^R



^0,1

 

^{ 0,1}



^.

(f) _x^{2 x} _y^{2 dA,}



R _ ^{R }



^1,2

 

^{ 0,1}



^.

5. Sketch the solid whose volume is given by the iterated integral.

(a) 4 2  .

1

0 1

0

dxdy y

 

^x^{ (b) 1}



²



^.

0 1

0

2

2 y dydx

 

^x ^

6. Find the volume of the solid that lies under the hyperbolic paraboloid

2

4 x2 y

z   and above the square ^R



^1,1

 

^{ 0,2}



^.

7. Find the volume of the solid enclosed by the surface z1x²ye^yand the planes z0, x 1, ^y0 and ^{y 1}.

8. Find the volume of the solid bounded by the surface ^{zx x2 y}and the planes x0, x 1, ^y0, ^y1 and z0.

9. Find the volume of the solid in the first octant bounded by the cylinder

9 y2

z  and the plane x2.

10. Find the volume of the solid enclosed by the paraboloid and the planes z 1,

1

x ,x1, ^y0 and ^y4. 11. Evaluate the double integral.

(a) 2 ₁dA^,

x y



D _ ^D



^{x,y|0x4 ,0y x}



. (b) x y dA,



D ^ D is bounded by ^{y x} and ^{y x2}. (c) ^{xcosy dA,}



D D is bounded by^y0, yx² and x1.

(d) ^{y2 dA,}



D D is the triangular region with vertices0,1,1,2 and

4,1 .

(e) 2x y dA,



D ^ D is bounded by the circle with center the origin and radius 2.

12. Find the volume of the given solid.

(a) Under the plane ^{3x2yz0}and above the region enclosed by the parabolas y x² and x y².

(b) Under the surfacezxy and above the triangle with vertices 1,1 ,

4,1 and 1,2 .

(c) Enclosed by the cylinders zx², ^{y x2} and the planes z0, ^y4 .

(d) Bounded by the cylinder x²y² 1 and the planes yz,x0,

0

z in the first octant.

13. Find the volume of the solid by subtracting two volumes.

(a) The solid enclosed by the parabolic cylinders^y1x2, ^{yx2 1} and the planes^{2x2yz100}.

(b) The solid under the plane z 3, above the plane z y, and between the parabolic cylinders y x² and y1x².

(c) The solid enclosed by the parabolic cylinder ^{y x2}and the planes

y

z3 , ^z2y.

14. Express D as a union of regions of type I or type II and evaluate the integral.

(a) x dA



D ² (b) ^{xy dA}



D

15. Use Property 11 to estimate the value of the integral.

(a) ^{x y dA}



S ^{4 2 2} , ^S

 x,y|x2y2 1,x0.

(b) ^{x y dA}



S ^{3 3} ,S 



0,1

 

 0,1



.

(c) ^{e dA}

S

y



x^{2 2} , S is the disk with center the origin and radius 2 1 .

16. A region is shown. Decide whether to use polar coordinates or rectangular coordinates and write fx ydA



R ^, as an iterated integral, where is an arbitrary continuous function on.

(a) (c)

(b) (d)

17. Sketch the region whose area is given by the integral and evaluate the integral.

(a)

 

^





2 7

4

rdrd . (b) _

 

 

² rdrd

0 cos 4

0

18. Evaluate the given integral by changing to polar coordinates.

(a) ^{xy dA}



D , where D is the disk with center the origin and radius 3.

(b)



^{x y}



^dA



R ^{cos 2  2} , where R is the region that lies above the x-axis within the circle ^{x2y2 9}.

(c) ^{x y dA}



R ^{4 2 2} , where ^R

 x,y|x2y2 4,x0.

(d) ^{ye dA}

R



x , where R is the region in the first quadrant enclosed by the circle x² y²25.

(e) _x^{y dA}



R ^tan1_^{ }_^{ , where R}

 x,y|1 x2y24 ,0 y x.

(f) ^{x dA}



D , where D is the region in the first quadrant that lies between the circles ^x2y24and x² y²2x.

19. Use a double integral to find the area of the region.

(a) The region enclosed by the curve r43cos .

(b) The region within both of the circlesrcosandrsin.

(c) The region inside the circle r 4sin and outside the circle r 2. 20. Use polar coordinates to find the volume of the given solid.

(a) Under the paraboloid ^{zx2 y2}and above the disk^x2y29. (b) Inside the sphere^{x2y2z2 16} and outside the cylinder ^x2y24. (c) Bounded by the paraboloids z3x²3y²and z4x²y²

21. Evaluate the iterated integral.

(a)

  

^

 1

0 1

1 0

4

x

x xy

dzdydx

z (c)

  

^{2 }

0 1

0 1 2

0

cos



dydxdz z

x

x .

(b)

  

^{3 }

0 1

0 1 2

0

dxdzdy ze^y

z

. (d)

  



 

2

1 2

1 y e2

e

dzdxdy z

y x

22. Evaluate the triple integral.

(a) yz

 

x dV

E

cos ⁵



, whereE

 

x,y,z



|0x1 ,0yx,xz2x



.

(b) ^xydV

E



6 , where E lies under the plane ^{z1xy} and above the region in the xy-plane bounded by the curves^{y x},^y0 and x1.

(c) ^xydV



E , where E is the solid tetrahedron with vertices 0 ,0 ,0,

^1,0,0, 0,2,0 and 0,0,3 .

(d) ^{x e dV}

E y



2 , where E is s bounded by the parabolic cylinder z1 y²

and the planesz0, x1 and x 1.

(e) ^xydV



T , where T is the first-octant solid bounded by the coordinate planes and the upper half of the sphere ^{x2y2z2 4}.

23. Express the integral as an iterated integral fx y zdV

E

,



, in six different ways, where E is the solid bounded by the given surfaces.

(a) x²z² 4, ^y0 and ^y6. (b) ^{9x24y2z21}

24. The figure shows the region of integration for the integral  , ,  .

1

0 1 1

  

^0 x

y

dzdydx z

y x

f

Rewrite this integral as an equivalent iterated integral in the five other orders.

25. Sketch the solid whose volume is given by the integral and evaluate the integral.

(a) ² _.

0 2

0 9 2

  

^0



 rdzdrd

r (b) ^{sin .}

2

0 2

2 2

  

^{ } 1











 d d d

26. Evaluate using cylindrical coordinates

(a)



^,

T

dxdydz  , , |0 1,0 1 ²,0 4 ^{2 2} .







         

 x y z x y x z x y

T

(b)



2¹_ 2 ^, T

dxdydz y

x

 , , |0 9 ²,0 3,0 9 ^{2 2} .







         

 x y z x y y z x y

T

(c)



^sin



^{2 2}



^,

T

dxdydz y

x  , , |0 1,0 1 ²,0 2 .







       

 x y z x y x z

T

(d)



^{2 ,}

E

dV

x where E is the solid that lies within the cylinderx²y²1

above the planez0and below the cone ^{z2 4x24y2.}

(e)



^,

E

xdV where E is enclosed by the planesz0and ^{zx y3}and by the cylinder ^x2y24and ^x2y29.

27. Find the volume of the solid bounded above by the plane 2z4x, below by the xy-plane, and on the sides by the cylinder ^x2y22x.

28. Find the volume of the solid bounded above by ^{x2y2z225}and below by

2 1

2 

 x y

z .

29. Evaluate the integral by changing to spherical coordinates.

(a) ^.

3

3 9 2

9 2

2 2 2 2 9 2

  

0

















x

x

y x

dzdydx z

y x z

(b)



^,

T

dxdydz

 ^{, ,} ^{|0 1,0 1 2, 2 2 2}  ^{2 2} ^.







          

 x y z x y x x y z x y

T

(c)

 

²_ ¹²_ ²



^,

T

dxdydz z

y x

 , , |0 1,0 1 ²,0 1 ^{2 2} .







         

 x y z x y x z x y

T

(d)



^,

E

zdV where E lies between the spheres ^{x2y2z2 1} and

2 4

2 2y z 

x in the first octant.

(e)



^{2 ,}

E

dV

x where E is bounded by the xy-plane and the hemispheres

2

9 x2 z

y   and y 16x²z².

(f)



^,

E

xyzdV where E lies between the spheres ^2and ^{ 4}and above the cone

3

  .

30. Find the volume of the solid common to the spherea and the cone ^{ }.

Take ^



 



 

 2

,1

0 .

31. Find the volume of the solid enclosed by the surface^{ 1cos}.

32. Find the volume of the solid that lies within the sphere x²y²z² 4, above the xy-plane, and below the cone^{z x2 y2.}