Question

Suppose the function f is continuous on the closed interval [a b]. Under what conditions is the value of the definite integral of f on [a b] equal to the area of a plane region that includes the graph of f on [a b] as a boundary? Explain when the value of the definite integral is not equal to the measure of the area of such a plane region.

Answer 1

Weird question, but when the function is negative, you end up getting negative values (and area is always positive), so I suppose you could say whenever the function isn't negative.

Answer 2

@wio do you know under what condition is it? and how to prove?

Answer 3

Basically if the function is ever negative in the interval \([a,b]\).

Answer 4

i need some proving please