128 Ordinary Differential Equations
- Find the area and arc length of the ellipse x = 4 cost, y = 5 sin t.
- The curve traced out by a point P on the circumference of a circle of
radius R as the circle rolls along the x-axis is called a cycloid. If when
t = 0, the point P is at the origin, then the equation of the cycloid is
x = R(t - sint) , y = R(l - cost).
a. Find the area under one arch of the cycloid with R = 2.
b. Find the arc length of one a rch of the cycloid wit h R = 3.
4 7. The curve traced out by a point P on the circumference of a circle of
radius r which rolls around on the inside of a larger circle of radius R
is called a hypocycloid. If the larger circle has equation x^2 + y^2 = R^2
and if when t = 0 t he point P is at (R, 0) , then t he equation of the
hypocycloid is(R - r)t
x = ( R - r) cost + r cos ,
r(R - r)t
y = ( R - r) sin t -r sin.
rFind the area and a rc length of t he hypocycloid with R = 3 and r = 1.
- The curve traced out by a point P on the circumference of a circle of
radius r which rolls around the outside of a circle of radius R is call ed
an epicycloid. If t he la rger circle h as equation x^2 + y^2 = R^2 and if when
t = 0 the point P is at (R , 0), then the equation of the epicycloid is
(R + r)t
x = (R + r) cost - r cos ,
r(R + r)t
y = ( R + r) sin t -r sin.
rFind the area and arc length of the epicycloid with R = 3 and r = 1.- The involute of a circle is the curve traced out by a point P at the end
of a string which is being unwound tautly from about a circle. If t he
equation of the circle is x^2 + y^2 = R^2 and if when t = 0 the point P is
at (R, 0) , t hen the equa tion of the involute of the circle isx = R(cost + tsint), y = R(sint - tcost).
Find t he arc length of the involute of a circle with R = 2 for t = 0 to
t = 27r.- Find the a rc length of the epitrochoid
x = 3 cos 2e + 4 cos e, y = 3sin2B + 4sinB.