Intuitive explanation for why the definite integral gives the area between the function and the x-axis Could somebody please give an intuitive explanation for why the antiderivative of a function evaluated at $b$ minus the antiderivative of the function evaluated at $a$, where $b>a$, gives the area between the function and the $x$-axis between these two $x$ values.
It does not make much sense to me, could somebody please give an intuitive proof or intuitive explanation
 A: Chop up the interval $[a,b]$ into tiny subintervals $[x_i,x_{i+1}]$.  Clearly the total change $f(b) - f(a)$ is equal to the sum of all the little changes $f(x_{i+1}) - f(x_i)$.  But, $f(x_{i+1}) - f(x_i) \approx f'(x_i) (x_{i+1} - x_i)$.  Thus, $f(b) - f(a) \approx \sum_i f'(x_i)(x_{i+1} - x_i) \approx \int_a^b f'(x) \, dx$.  When we chop up $[a,b]$ more and more finely, the approximations get better and better, so by a limiting argument we discover that $f(b) - f(a) = \int_a^b f'(x) \, dx$.
A: Let's assume we have a velocity function $v(t)=10$. The integral of velocity is position. The integral is the sums of the y-values (velocity) at infinitesimally small $t$ intervals between $a$ and $b$.
Let's do an integral from $a=2$ to $b=4$:
$$
\int_2^4{10\mathrm{d}t}\\
=x(t)\Big{|}_2^4\\
=5t\Big{|}_2^4\\
=(5\times4)-(5\times2)\\
=10
$$
What does that 10 mean? It represents the change in position BETWEEN two limits $a$ and $b$. The reason that is important is that your definite integral with limits $a,b$ is essentially integral of v(t) from a to b equals antiderivative at b minus antiderivative at a.
Let $x=\int{v(t)}\mathrm{d}t=5t$ be our position function. If you evaluate your antiderivative at $b$, then you are integrating the entire domain of the function up to $b$... for our position and velocity functions, the lower end of the domain is the beginning of time!
When you subtract the antiderivative evaluated at $a$, you chop off everything that came before $a$, so you're only calculating how much your position changed over your particular interval.
