Integration and Accumulation of Change

If $g(x) = \int_{a}^{x} f(t) dt$, then according to the Fundamental Theorem of Calculus, what is $g'(x)$?

Given $g(x) = \int_{0}^{x} \sqrt{8 + \cos(t)} dt$, find $g'(0)$.

Given $g(x) = \int_{1}^{x} (5t^2 + 2t) dt$, find $g'(3)$.

If $F(x) = \int_{3}^{x^2} (t + 4) dt$, what is $F'(x)$?

Let $F(x) = \int_{3}^{\sin(x)} t^2 dt$. Find $F'(x)$.

Given $F(x) = \int_{x^2}^{x^3} \sin(t^2) dt$, find $F'(x)$.

Suppose water is flowing into a tank at a rate of $r(t) = 3t^2 + 2t$ gallons per minute, where $t$ is measured in minutes. If the tank initially contains 5 gallons of water, how much water is in the tank after 2 minutes?

The rate of sales of a new product is modeled by the function $S(t)$, where $t$ is the time in weeks since the product was introduced. What does $\int_{0}^{5} S(t) dt$ represent?

A function $f(t)$ is defined as follows: $f(t) = t$ for $0 \le t < 2$ and $f(t) = 4 - t$ for $2 \le t \le 4$. Find the total accumulation from $t = 0$ to $t = 4$, i.e., $\int_{0}^{4} f(t) dt$.

A particle's velocity is given by the piecewise function: $v(t) = t^2$ for $0 \le t < 1$ and $v(t) = 2-t$ for $1 \le t \le 3$. What is the total distance traveled by the particle from $t=0$ to $t=3$?