One part of the fundamental theorem of calculus is that
$$F(x)=\int_a^x f(t)\;dt\tag1$$
However,
$$\int_a^b f(t)\;dt=F(b)-F(a)\tag2$$
So my first question is why doesn’t equation 1 take the form of $\int_a^x f(t)\;dt=F(x)-F(a)$? Where did $F(a)$ disappear to?
Also, whenever you see an integral in the form of $F(x)=\int_x^a f(t)\;dt$,
why is it that you must always change it to $F(x)=-\int_a^x f(t)\;dt$? That is, why is it necessary for $x$ to be the upper limit and not the lower? I know that it’s written in the fundamental theorem of calculus as the upper limit, but why?
Best Answer
You haven't actually stated the theorem. This is where being precise is important.
The first part says:
If $f$ is continuous on $[a,b]$ then the function DEFINED by:
$$F(x)\colon = \int_a^x f(t) dt$$
for $x\in [a,b]$
is differentiable on $(a,b)$ and $F'(x)=f(x)$.
The second part says:
If $F(x)$ is differentiable on $(a,b)$ with derivative $f(x)$ and $f(x)$ is continuous on $[a,b]$ then:
$$\int_a^b f(t) dt = F(b)-F(a)$$
Just stating theorems precisely will help clear up misconceptions.