120 Ordinary Differential EquationsIn what follows we summarize some results and formulas from calculus for
computing area, arc length , and volume when the curves involved are defined
in rectangular coordinates, in polar coordinates, and parametrically.
Formulas Involving Curves Defined in Rectangular Coordinates
The Area Under a Curve
Let y = f(x) b e a co ntinuous, nonnegative function (f(x) 2 0) on the
interval [ a, b]. The area, A , of the region in the xy-plane bounded above by
the curve y = f(x), bounded below by the x-axis (y = 0), and bounded by
the vertical lines x = a and x = b is
A= lb f (x) dx.
The Area Between Two Curves
Let y = f(x) and y = g(x) b e continuous functions on the interval [a, b]
with the property t hat f(x) 2 g(x) for all x E [a, b]. The area bounded above
by the curve y = f(x), bounded below by the curve y = g(x), and bounded
by the vertical lines x = a and x = b is
A= lb [f(x) - g(x)] dx.
Arc Length
If y = f(x) has a continuous first derivative, f'(x), on the interval [a, b],
then the arc length of the curve y = f(x) from a to bis
s =lb Ji+ [f'(x)]2 dx.
Areas of Surfaces of Revolution
If y = f(x) has a continuous first derivative, f'(x), on the interval [a , b],
then
( 1) the area of t he surface generated by revolving the curve y = f ( x) from a
to b about the x-axis is
Bx = 27r lb lf(x) I Jl + [!' (x )]2 dx
and
(2) the a rea of t he surface generated by revolving about the y-axis the curve
y = f(x) from a to b where 0 ~a~ bis
Sy= 27r lb xJl + [f'(x)]2 dx.