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

CHAPTER 10 REVIEW 0 35
the length of the orbit is
L = 12
,. Jr 2 + (dr/d0)2 d(J = a(l- e 2 ) 12,. .Jl(t e 2 + 2 ~)~sO dO:::::: 3.6 x 10 8 km
0 0 +ecos
This seems reasonable, since ,Mercury's orbit is nearly circular, and the circumference of a circle of radius a
is 21ra :::::: 3.6 x 10 8 km.·
10 Review
CONCEPT CHECK
1. (a) A parametric curve is a set of points of the form (x, y) = (J(t), g(t)), where f and g are continuous functions of a
variable t.
(b) Sketching a parametric curve, like sketching the.graph of a function, is difficult to do in general. We can plot points on the
curve by finding f(t) and g(t) for various values oft, either by hand or with a calculator or computer. Sometimes, when
f and g are given by formulas, we can eliminate t from the equations x = j(t) andy = g(t) to get a Cartesian equation
relating x andy. It may be easier to graph that equation than to work with the original formulas for x andy in terms oft.
2. (a) You can find : as a function oft by ca lcu lati ng~~ = ~~J~~ [if dxjdt f: 0].
(b) Calculate the area as I: y dx = I: g(t) j'(t)dt [or I; g(t) !' (t)dt ifthe leftmost point is (!({3), g({J)) ~ther
than (f(a),g(a))].
3. (a) L = I: J(dxjdt)2 + (dyjdt)2 dt =I: vlf'(t))2 + [.g'(t))2 dt
(b) s =:: I: 27ryJ(dxfdt) 2 + (dyfdt) 2 dt =I: 21rg(t) v lf'(t)F + [g'(t)J2 dt .
4. (a) See Figure 5 in Section 10.3.
(b) x = rcosO, y = r sinO
(c) To find a polar representation (r, 0) with r ;?. 0 and 0 s 0 < 27r, first calculate r = ·Jx 2 + y2. TI1en 0 is specified by
cosO= xjr and sinO= yjr.
d dy !:.._ (y) !:.._ ( r ::;in 8)
5. (a) Calculate Jx = ~ = ~ = ..::::llJ:;:-- -
dO dO (x) d() (rcosB)
(b) calculate A= I: !r 2 dO= I: tlfCOW d8
(~) sin(} + r cos(}
dr) . , wherer = f(O).
( dO cos () - r sm (}
(c) L =I: J(dx/d8.)2 + (dy/d0) 2 dO= I: Jr 2 + (dr/d0) 2 dO= .r: J[f(0))2 + [!'(0)]2 dO
6. (a) A parabola is a set of points in a plane whose distances from a fixed point F (t11e focus) and a fixed line l (the 'directrix)
are equal.
(b) x 2 = 4py; y 2 = 4px
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