Question

In lecture 20, when the professor is proving the fundamental theorem of calculus 2 starting at around 30:00, is it okay to have x in function in ftc 2 as well as in the upper limit of the integrand? I thought you needed some other variable like t in the function and x in the integrand.

Answer 1

The function displayed as a curve on the graph is f(x), and there's no problem there. The place where we need the dummy variable t is in G(x), where it serves to avoid using the same variable in two inconsistent ways (as the independent variable and as the upper limit of the integral).

Answer 2

Okay, but when the professor is draws an arrow to the part below the curve f(x) from a to x and calls it G(x), according to FTC 2, which he points to right after, he would be saying \[G(x) = \int\limits_{a}^{x} f(x) dx\]because f(x) is the curve on the graph?
I somewhat understand how he's right because of his explanation around a minute or two prior; he says that G'(x) = f(x), so that makes sense.

Answer 3

It is indeed the area under a curve that represents f(x), but the area is equal to the integral from a to x of f(t)dt. In substituting the variable t, we aren't changing the quantity being measured, we're just inserting a placeholder that prevents us from lapsing into a logical inconsistency when using x as the upper bound of the integral.

It might help to remember that we're speaking loosely when we say f(x) is a function. Strictly speaking, f is a function, and f(x) is the result of applying that function to x. When we apply that function to t, it does the same thing to t as it would do to x, so we can determine the area under the curve representing f(x) in terms of f(t). That's why Prof. Jerison was able to say G(x), which is defined in terms of f(t), provides the area under the curve f(x). I think he was trying to make this explicit when he pointed to the area under the curve on his diagram and then walked over to the board where he defined G(x), pointing to that function and saying the area is G(x).

The variable t is a dummy variable. Its use doesn't alter the calculation. It merely allows us to sidestep problems that can arise from using the same variable in two inconsistent ways.

Answer 4

Ahh, now I understand, thank you so much! The way you explained f as the function and f(x) is applying the function to x cleared it up. Thanks again!