Thomas Calculus 13th [Solutions]

Section 13.2 Integrals of Vector Functions; Projectile Motion 933
35-40. Example CAS commands:
Mathematica: (assigned functions, parameters, and intervals will vary)
The x-y-z components for the curve are entered as a list of functions of t. The unit vectors i, j, k are not
inserted.
If a graph is too small, highlight it and drag out a corner or side to make it larger.
Only the components of r[t] and values for t0, tmin, and tmax require alteration for each problem.
Clear[r, v, t, x, y, z]
r[t _] { Sin[t] t Cos[t], Cos[t] t Sin[t], t^2}
t0 3 /2; tmin 0; tmax 6 ;
ParametricPlot3D[Evaluate[r[t]], {t, tmin, tmax}, AxesLabel
v[t _]
r'[t]
tanline[t _] v[t0]t r[t0]
{x, y, z}];
ParametricPlot3D[Evaluate[{r[t], tanline[t]}], {t, tmin, tmax}, AxesLabel {x, y, z}];
For 39 and 40, the curve can be defined as a function of t, a, and b. Leave a space between a and t and b and t.
Clear[r, v, t, x, y, z, a, b]
r[t _,a_,b_]: {Cos[a t], Sin[a t], b t}
t0 3 /2; tmin 0; tmax 4 ;
v[t _,a_,b_] D[r[t, a, b], t]
tanline[t _,a_,b_] v[t0, a, b]t r[t0, a, b]
pa1 ParametricPlot3D[Evaluate[{r[t, 1, 1], tanline[t, 1,1]}], {t,tmin, tmax}, AxesLabel {x, y, z}];
pa2 ParametricPlot3D[Evaluate[{r[t, 2, 1], tanline[t, 2, 1]}], {t,tmin, tmax}, AxesLabel {x, y, z}];
pa4 ParametricPlot3D[Evaluate[{r[t, 4, 1], tanline[t, 4, 1]}], {t,tmin, tmax}, AxesLabel {x, y, z}];
pa6 ParametricPlot3D[Evaluate[{r[t, 6, 1], tanline[t, 6,1]}], {t,tmin, tmax}, AxesLabel {x, y, z}];
Show[GraphicsRow[{pa1, pa2, pa4, pa6}]]
13.2 INTEGRALS OF VECTOR FUNCTIONS; PROJECTILE MOTION
1.
1 4 1 2 1
3 1
t 7 ( t 1) dt t 7t t t 1 7 3
0 4 0
0
2
0
4 2
i j k i j k i j k
2 2
2
3/2
2
1
2
2. (6 6 t) i 3 t j
4
k dt 6t 3t i 2t j 4t
k 3i 4 2 2 j 2k
1
2
t
1 1 1
3.
/4 2 /4 /4 /4 2 2
(sin
/4 t ) (1 cos t ) sec t dt cos t sin tan 2
/4 t t /4 t /4 2
i j k i j k j k
4.
5.
/3 /3
(sec t tan t) i (tan t) j (2sin t cos t) k dt (sect tan t) i
0 0
(tan t) j (sin 2 t)
k dt
/3 /3 /3
sec t i ln(cos t) j
1 cos 2 t k i (ln 2) j
3
k
0 0 2 0
4
4
1 1 1 4 4 1
4
dt
1 5 2 ln t
1 ln (5 t ) 1 2
ln t
t t t
(ln 4) (ln 4) (ln 2)
1
i j k i j k i j k
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