Applications of the Initial Value Problem y' = f(x, y); y(c) = d 121
Volumes of Solids of Revolution
Let J(x) and g(x) be continuous functions on the interval [a, b] with the
property that f(x) ;::: g(x) ;::: 0 for all x E [a, b]. The volume of the solidgenerated by revolving the region bounded by the curves y = f(x), y = g(x),
x =a, and x = b
(1) about the x-axis isand
(2) about the y-axis for 0 :::; a :::; b isVy= 27r 1b x [f(x) - g(x)] dx.
Formulas Involving Curves Defined in Polar Coordinates
Area
Let r = J(B) be a continuous, nonnegative function on the interval [a,,B]
where 0 < {3-a:::; 27r. The area of the region bounded by the curves r = J(B),
e = a ) and e = {3 is
The Area Between Two Curves
Let r = J(B) and r = g(B) be continuous functions on the interval [a, ,B]
where 0 < {3-a :::; 27r. And let f and g have the property that 0:::; f(B) :::; g(B)
for all e in [a, ,BJ. The area of the region bounded by the curves r = f ( B),
r = g(B), e =a, and e = {3 is
Arc Length
If r = f(B) has a continuous first derivative on the interval [a, {3], then the