When we defined the definite integral, we lifted the requirement that f(x) be nonnegative. But how do we interpret “the area under the curve” when f(x) is negative? Net Signed Area Let us return to the Riemann sum. Consider, for example, the function f(x)=2−2x2 (shown in (...
nite theory to the Friedrich's extension of the classic Bessel di¤erential expression on (0; 1) to obtain new integral inequalities; in fact, the …rst left-de…nite theory implies the classic 1920 Hardy inequality while the higher-order left-de…nite spaces produce new generalizations, ...
We focus on definite integrals of continuous functions in one variable over closed intervals. In particular, we consider expressions given by the following syntax: e := v | c | e1 op e2 | f (e) | Deriv(e, v) | Integral(e, v, a, b) Here v is a variable; c is a constant (...