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

268 0 CHAPTER 15 MULTIPLE INTEGRALS
2 2 2 2 2 2
13. z = f(x, y) =e-x - v , f.,= -2xe-x - v , fv = -2ye-x - v . Then
A(S)= II v(- 2xe- xLv 2 )2+(- 2ye- x 2 -v 2 )2+ 1dA =
x2+y2_54
Converting to polar coordinates we have
If .j4(x2-t-y2)e- 2(x 2 +v 2 l-t- ldA.
x2+1125-1
A(S) = J:" .{ 0
2
J 4r 2 e-' 2 r 2 + 1 r dr d(J = ;;11: dl1 .{ 0
2
r V 4T 2 e- 2 '' 2 + 1 dr
= 27!' .r; r V 4r 2 e- 2 r 2 + 1 dr ~ 13.9783 using a calculator.
15. (a) The midpoints of the four squares are (i 1 i ), (i 1 %) , (%, ~)~and ( i 1 ~). Here f(x 1 y) = x 2 + y 2 , so the Midpoint Rule
gives
A(S) = JJD .j[fx(X 1 y)F + [fv(x, y)J2 + 1 dA = jjD .j(2x) 2 + (2y) 2 + 1 dA
~ ~ ( J[2(~)J 2 + (2(iW + 1 + V[2(~) J 2 + [2(t~] 2 + 1
+ V[2(~)] 2 + [2(i)] 2 + 1 + V[2(%W + [2(~)] 2 + 1)
= ~ ( .ft + 2 JJ + /¥) ~ 1.8279
(b) A CAS estimates the integral to be A(S) = J[D .)1 + (2x ) 2 + (2y )2 dA = .1; J 0
1
.)1 + 4x 2 + 4y 2 dy dx ~ 1.8616.
This agrees with the Midpoint estimate only in the first decimal place.
17. z = 1 + 2x + 3y -t--4y 2 , so
4
·A(S) = 11 1 + ( ~~r + (~~r dA = 1.f Using a CAS, we have J 1
4
45 'P4 15 ln 11 J5 + 3 v'70
or- V J. '% + -
8 16 3 J5 +. v'70 .
4
v 1 +4+(3 + 8y)2dydx = 1
\
1
V 14+48y+64y2dydx.
I~ .)14 + 48y + 64y 2 dy dx = ¥ .Jl4 + ~ ln(ll J5 + 3 .Jl4J5) - t~ ln(3 J5 + v'i4 J5)
19: f(x, y) = 1 + x 2 y 2 => f, = 2xy 2 , fv = 2x 2 y. We use a CAS (with precision reduced to five significant digits, to speed
up the calculation) to estimate the integral
A(S) =
1
V
!1~ 1!1~ '
fi + F~ + 1 dy dx = ..j 4x 2 y 4 + 4x 4 y 2 + 1 dy dx, and find that A(S) ~ 3.3213.
- 1 - vh-x.2 . - 1 -~ ·
21. Here z = f(x , y) = ax+ fnJ + c, f.,(x 1 y) =a, / 11 (x, y) = b, so
I
A(S) = JJD ,Ja 2 + bz + 1 dA = ,ja2 + b 2 + 1JJD dA = ,Ja 2 + b 2 + 1 A( D).
23. If we project the surface onto the xz-plane, then the surface lies "above" the,' disk x 2 + z 2 ~ 25 in the x z-plane.
We have y = f(x, z) = x 2 + z 2 and, adapting Formula 2, the area of the surface is
A(S) = J{ .j(f,.(x,z)]2+[fz(x,z)]2+1dA = JJ ,J4x 2 +4z 2 +1dA
,;2 +z2 .$25 ,2 +=2 .$25
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