# Integral de superficie

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1030

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CHAPTER 15 MULTIPLE INTEGRALS

Because of the symmetry of E and about the xz-plane, we can immediately say that Mxz 0 and, therefore, y 0. The other moments are Myz

yyy x
E

dV
1
2

y y y
1 y
2

1

1

x

0

x dz dx dy
1 1

y y
1

1

y

x 2 dx dy

y

x3 3 y

x 1

dy
x y2

2 3

y

1

0

1

y 6 dy

2 3
x

y7 7

1

0

47

Mxy

yyy z
E 1

dV
1
2

y y y
1 y
2

1

1

0

z dz dx dy

y y
1 1

y

z2 2

z x

dx dy
z 0

2

y y
1

1

1
2

y

x 2 dx dy

y 3

0

1

y 6 dy

2 7

Therefore, the center of mass is x, y, z Myz Mxz Mxy , , m m m
5 ( 5 , 0, 14 ) 7

|||| 15.7

Exercises
10. 11.

1. Evaluate the integral in Example 1, integrating ﬁrst with

xxxEy dV ,
z

respect to z , then x, and then y.
2. Evaluate the integral xxxE xz

where E is bounded by the planes x 0, and 2x 2y z 4

0, y

0,

y 3 dV , where 1, 0 y 2, 0 z 1

xxxE xy dV , xxxE xz dV ,
z 1

E

x, y, z

1

x

where E is the solid tetrahedron with vertices 0, 0, 0 , 1, 0, 0 , 0, 2, 0 , and 0, 0, 3

12.

using three different orders of integration.
3–6||||
1

where E is the solid tetrahedron with vertices 0, 0, 0 , 0, 1, 0 , 1, 1, 0 , and 0, 1, 1 where E is bounded by the parabolic cylinder y 2 and the planes z 0, x 1, and x 1

Evaluate the iterated integral.
z x z

13.

xxxE x 2e y dV , xxxE
y

3. 5.
s

y0 y0 y0
3 1

6xz dy dx dz ze y dx dz dy
s s s

4. 6.
s

y0 yx y0 2xyz dz dy dx y0 y0 y0 ze
s s

1

2x

y

14.15.

y0 y0 y0
s

s1 z 2

1

z

y

y2

x 2y dV , where E is bounded by the parabolic cylinder x 2 and the planes x z, x y, and z 0 4y 2 4z 2 9 and
s s

dx dy dz
s s s

xxxE x dV , where E is bounded by the paraboloid x
and the plane x 4

s

7–16

||||

Evaluate the triple integral.

16.

where E is bounded by the cylinder y 2 z 2 the planes x 0, y 3x, and z 0 inthe ﬁrst octant

xxxE z dV ,
s s

7. 8. 9.

xxxE 2x dV , where
E E

s

s

s

s

s

s

s

s

{ x, y, z

0

y

2, 0

x y

s4

y 2, 0 z 2x

z

y}
17–20
||||

xxxE yz cos x 5 xxxE 6xy dV ,

dV , where x, y, z 0 x 1, 0

Use a triple integral to ﬁnd the volume of the given solid. y z 4 x 2 and the planes 9

x, x

17. The tetrahedron enclosed by thecoordinate planes and the

where E lies under the plane z 1 x y and above the region in the xy-plane bounded by the curves y sx, y 0, and x 1

plane 2x z 0, z

18. The solid bounded by the cylinder y

4, and y

SECTION 15.7 TRIPLE INTEGRALS

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1031

19. The solid enclosed by the cylinder x 2

y2 y2
s

9 and the planes z 2 and the
s s s

31. The ﬁgure shows the region ofintegration for the integral

y

z

5 and z 16
s s

1

20. The solid enclosed by the paraboloid x

plane x
s s

y0 ysx y0

1

1

1 y

f x, y, z dz dy dx

s

s

s

s

21. (a) Express the volume of the wedge in the ﬁrst octant that is

Rewrite this integral as an equivalent iterated integral in the ﬁve other orders.
z 1

CAS

cut from the cylinder y 2 z 2 1by the planes y x and x 1 as a triple integral. (b) Use either the Table of Integrals (on the back Reference Pages) or a computer algebra system to ﬁnd the exact value of the triple integral in part (a).
22. (a) In the Midpoint Rule for triple integrals we use a triple

z=1-y

CAS

Riemann sum to approximate a triple integral over a box B, where f x, y, z is evaluated at the center xi , yj ,zk of the box Bijk. Use the Midpoint Rule to estimate 2 2 2 xxxB e x y z dV , where B is the cube deﬁned by 0 x 1, 0 y 1, 0 z 1. Divide B into eight cubes of equal size. (b) Use a computer algebra system to approximate the integral in part (a) correct to two decimal places. Compare with the answer to part (a). Use the Midpoint Rule for triple integrals (Exercise 22) to estimate the value of the...