Applications of the Initial Value Problem y' = f(x, y); y(c) = d 127- The spiral of Archimedes is given by the equation r = a8 where a is a
constant. For a = 3 find the area inside the spiral of Archimedes from
8 = 0 to 8 = 27r. For a = 4 find the arc length of the spiral of Archimedes
from 8 = 0 to 8 = 7r.
36. Find the area inside the logarithmic spiral r = 2e3 8 from 8 =Oto 8 = 7r.
Find the arc length of this spiral.37. Find the area and arc length of the cardioid r = 2 + 2 cos 8.
(See Figure 3.2.)- Find the area and arc length of the limac;on r = 3 - 2 sin 8.
(See Figure 3.3.) - Find the area and arc length of the rose curve r = 4 cos 38.
(See Figure 3.4.) - Find the area and arc length of the lemniscate r^2 = 9 sin 2B.
(See Figure 3.6.) - Find the area between the two loops of the limac;on r = 1 - 2 cos 8.
Find the arc length of each loop of the limac;on. (See Figure 3.1.) - Find the area inside both lemniscates r^2 = cos 28 and r^2 = sin 2B.
- Find the area between the two limac;ons r = 5 + 3 cos 8 and r = 2 -sin 8.
- About 1638 , Rene Descartes (1596-1650) sent the equation x^3 +y^3 = 3xy
to Pierre Fermat (1601-1665) and challenged him to determine the tan-
gent line to the curve at any point. (Can you determine the tan-
gent line at any point?) The graph of the equation x^3 + y^3 = 3xy
is call ed the folium of Descartes. A sketch of the graph is shown in
Figure 3.7. The equation may be rewritten in polar coordinates as
r = 3 sin 8 cos 8 I ( cos^3 8 + sin^3 8). Find the area and arc length of the
loop ("leaf") in the folium of Descartes.
3 y
2 x3+y3=3xyx-3 3
Figure 3.7 Graph of the Folium of Descartes