Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS 0 251
31. The solid lies below the surface z = 2 + x 2 + (y- 2? and above the plane z = 1 for -1 ~ x ~ 1, 0 ~ y ~ 4. The volume
of the solid is the difference in volumes between the solid that lies under z = 2 +x 2 + (y- 2) 2 over the rectangle
R = (-1, 1] x (0, 4) and the solid that lies under z = 1 over R.
V = J; J~ 1 (2 + x 2 + (y :._ 2?J dxdy- f 0
4
f~ 1 (1) dxdy = f 0
4
[2x + 4x
3
+ x(y - 2?J:: ~ 1
dy- t 1
dx J;dy
= J; [(2 + ~ + (y - 2) 2 )- ( -2- ~- (y- 2) 2 )] dy- [x]~ 1 (y]~
4
= f 0
(lf + 2(y- 2) 2 ) dy- (1 - (-1)][4- OJ = ( 1 4
3 y + t(Y- 2?)~- (2)(4)
= [e; + ¥)- (o - nJ 1 -8 = ¥-8 = ¥
33. In Maple, we can calculate the integral by defining the integrand as f
and then using the command int ( int ( f, x=O .. 1) , y=O . . 1) ; .
In Mathematics, we can use the command
Int egrate[f,{x,0,1}, {y,0,1}]
We find that J n x 5 y 3 e