So I have a "noob" understanding of calculus. In an integral, I forget, while I realize the limits of the two, what value does the upper and lower limit have in the equation to finding the $d/dx$ or derivative? More specifically I have a small example:
$y = mx + b$
$y = f(x)$
$f(x) = \dfrac{8}{5} x + 2$
The upper limit is $5$ and the lower limit is $0$. The integral is $$\int_0^5 \frac{8}{5} x + 2.$$ I know $$\frac{x^{n+1}}{n+1}$$ is the derivative. But I don't see where the upper and lower limit is being used! Please explain.
The integral of a function $f(x)$ from $x = a$ to $x = b$, written as $$\int_a^bf(x)dx$$ is the signed area between the graph of the function $f(x)$ and the $x$-axis, from $x = a$ to $x = b$, as can be seen in this image.
One half of the fundamental theorem of calculus says that if $F(x)$ is an antiderivative of $f(x)$ (i.e. $F'(x) = f(x)$) then $$\int_a^bf(x)dx = F(b) - F(a).$$ So, when presented with such an integral, you must first find an antiderivate (presumably using the laws you have learnt/are learning) then evaluate the antiderivative at the upper limit and subtract from that the value of the antiderivative at the lower limit.