1550078481-Ordinary_Differential_Equations__Roberts_

122 Ordinary Differential Equations

Formulas Involving Curves Defined Parametrically

Area

If y( x) is a nonnegative, continuous function of x on the interval [ a, b], if x

and y are defined parametrically by

x = f(t), y = g(t) for ti ::; t ::; t2

where f(ti) =a and f(t 2 ) = b, and if f'(t) and g(t) are both continuous on
the interval [t 1 , t 2 ], then the definite integral of y(x) on the interval [a, b]- the
area under the curve y(x) over [a, b]- is

l

b 1 t2

y(x)dx = g(t)f'(t)dt.

a ti

Arc Length

If a curve is defined parametrically on the interval [a, b] by x = f(t), y = g(t)

for t 1 ::; t ::; t2 where f(t 1 ) = a and f(t2) = b and if f'(t) and g'(t) are

continuous on t he interval [t 1 , t 2 ], then t he arc length of the curve over [a, b]

is

1

t2 1 t2

s = ..j(dx/dt)^2 + (dy/dt)^2 dt = J[!'(t)]^2 + [g'(t)]2 dt.

ti t1

EXAMPLE 2 Numerical Calculation of Arc Length,

Surface Area, and Volume

a. Find the arc length of the semi-circle y(x) = Jg - x^2 over the interval

[l , 2].

b. Find the area of the surface generated by revolving the given arc of the

semi-circle over t he interval [l , 2] about the y-axis.

c. Find the volume of the solid generated by revolving the region bounded by

the given arc of the semi-circle, the h orizontal line y = J5, and the vertical

line x = 1 about the y-axis.