If 5. x = 3 sin 2t, y = 3 cos 2t, (0 t /3), then x 2 + y 2 = 9. This is part of a circle.
y t=0 x 2 +y 2 =9 x t= 3
Fig. 2-5
7. If x = 3 sin t, y = 4 cos t, (-1 t 1), then y2 x2 + = 1. This is an ellipse. 9 16
y t=0 x 2 y2 9 + 16 =1
x
t=-1 t=1
Fig. 2-7
9. If x = cos3 t, y = sin3 t, (0 t 2 ), then x 2/3 + y 2/3 = 1. This is an astroid.
y t=/2 x 2/3 +y 2/3 =1 t= t=0 t=2
t=3/2
Fig. 2-9
21.
y = x 3 , 0 x 1. ds = 1 + 9x 4 d x. The area of the surface of rotation about the x-axis is
1
S = 2
0
x 3 1 + 9x 4 d x u du =
Let u = 1 + 9x 4 du = 36x 3 d x
=
18
10 1
(103/2 - 1) sq. units. 27
23. If y = x 3/2 , 0 x 1, is rotated about the y-axis, the surface area generated is
1
S = 2
0
x 1+
9x dx 4
Let u = 1 + du = 9 dx 4
9x 4
32 81 32 = 81 = = 81
13/4 64 1
(u - 1) u du
13/4 1
2 5/2 2 3/2 u - u 5 3 (13/4)5/2 5 -1 -
(13/4)3/2 - 1 3
sq. units.
25. If y = sin x, 0 x , is rotated about the x-axis, the surface area generated is
S = 2
0 1
sin x 1 + cos2 d x 1 + u 2 du
-1 /4
Let u = cos x du = - sin x d x
= 2 = 2
-/4
Let u = tan du = sec2 d
/4 0 /4 0
sec3 d = 4
sec3 d
= 2 sec tan + ln | sec + tan | = 2 2 + ln(1 + 2) sq. units.
27. For y =
x3 1 + , 1 x 4, we have 12 x 1 x2 + 2 d x. ds = 4 x The surface generated by rotating the curve about the y-axis has area
4
S = 2
1
x
1 x2 + 2 4 x
dx
= 2 = 2
4 x4 + ln |x| 16 1 255 + ln 4 sq. units. 16
3. If x = 1/t, y = t - 1, (0 < t < 4), then y =
y 1 4 ,3 t=4
1 - 1. This is part of a hyperbola. x
1 y= x - 1 t=1 x y=-1
Fig. 2-3