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Currently reading "Essential Mathematics for Economic Analysis" by Sydsaeter and Hammond. I have a question from the book (page 313 if anyone cares).

Assume [tex]f(x)=x^{2}[/tex]

Also assume that [tex] F(x)=\frac{1}{3}x^{3}[/tex] is an indefinite inegral of [tex]f(x)[/tex].

Now I want to calculate the area under f(x) in interval [0,1]. I do this by calculating [F(1)-F(0)] and get 1/3 as answer. The authors then say that 1/3 is a "reasonable" answer. From this comment I assume that the area is only approximated by 1/3.

Why is this? Why does the F(1)-F(0) not give me exact answer for the area under f(x)=x^2.

I have two versions as an answer.

First is that any indefinite integral that we reconstruct from f(x) will always be just an approximation. I also think that in case of linear functions the indefinite integral will give the precise area. Correct?

Second is that there does exist an indefinite integral of f(x) that gives the perfect true area under f(x)-x^2, but it has more terms involving x that we cannot reconstruct from f(x).