Fundamental Theorem of Calculus problem I don't understand the intuition behind this.  Why can we just plug in $x$ for $t$ here and that gives us the result?  I thought I was understanding the Fundamental Theorem of Calculus, but I don't see how it applies here.  I thought the Theorem mainly stated that the area under a function can be found by taking the the value of the anti derivative over the specified interval. It doesn't make sense to me why we just plug in $x$ and voila that's our answer. 
$$\frac {d}{dx} \int_{a}^{x} (t^3 + 1) \ dt = x^3 +  1$$
 A: Since the function $\,t^3+1\,$ is continuous (and derivable) everywhere, it has a primitive function $\,G(t)\,$in any finite interval. Using the FTC , write
$$\int_a^x(t^3+1)dt=G(x)-G(a)\\\Longrightarrow \frac{d}{dx}{\left(\int_a^x(t^3+1)dt\right)}=\frac{d}{dx}{(G(x)-G(a))}=G'(x)=x^3+1$$
A: The Fundamental Theorem of Calculus doesn't talk about geometrical results, but about the "fundamental" relation between the operation of integration and that of differentiation. Namely, it says the following:

THEOREM. Let $f$ be a continuous over $[a,b]$. Define $F$ on $[a,b]$ by
$$F(x)=\int_a^x f(t) dt$$
Then $F$ is differentiable, and $F'(x)=f(x)$.

The corollary is

COROLLARY Let $f$ be continuous over $[a,b]$ and $f=g'$ for some $g$.
Then
$$\int_a^b f(t)dt=g(b)-g(a)$$

Note we can find this reversed in the books (One is the theorem and the other the corollary, or one is called FTC 1 and the other FTC 2). I recommend you read these two questions some users already asked about FTC:

*

*Understanding the relation between differentiation and integration.


*How do explain FTC to my teacher?
