Visual Calculus - Areas bounded by polar curves

Definition. Suppose that r = f(q) is a positive continuous function defined on the closed interval [a, b] where 0 < b - a < 2p. The area bounded by the graph of r = f(q), q = a, and q = b is

Discussion [Using Flash] [Using Java]

Exercises:

Find the area bounded by the graph of f(q) = 2 + sin(q).
Solution [Using MathView]
Find the area bounded by the graph of f(q) = 4 sin(q).
Solution [Using Flash] [Using Java]
Find the area bounded by one loop of the graph of f(q) = sin(3q).
Solution [Using Flash] [Using Java]
Find the area bounded by the small loop of the graph of f(q) = 2 + 3 sin(q).
Solution [Using Flash] [Using Java]

Comment. In the previous exercise, the function f was negative on the domain which defined the small loop. The definition for area given above also works for negative functions. However, extreme care must be taken when finding areas defined by functions which have both positive and negative images.

Examples:

f(q) = sin(3q).
Calculations [Using MathView]
f(q) = 2 + 3 sin(q).
Calculations [Using MathView]