Applications of Integration

What is the formula to calculate the average value of a function $f(x)$ over the interval $[a, b]$?

Find the average value of the function $f(x) = x^2$ on the interval $[0, 3]$.

A particle moves along the x-axis with velocity given by $v(t) = 3t^2 + 2t$. What is the displacement of the particle from $t = 1$ to $t = 3$?

Find the area of the region enclosed by the curves $y = x^2$ and $y = 4$.

Find the area of the region enclosed by the curves $y = x^2$ and $y = 2x$.

A particle's acceleration is given by $a(t) = 6t$, with initial velocity $v(0) = 5$ and initial position $s(0) = -3$. Find the position of the particle at $t = 2$.

Water leaks from a tank at a rate of $R(t) = 5e^{-t/20}$ gallons per hour, where $t$ is measured in hours. How much water leaks out of the tank during the first 10 hours?

The rate of production at a factory is given by $P(t) = 100 + 10t - 0.1t^2$ units per day, where $t$ is the number of days since the beginning of the year. What is the total number of units produced during the first 30 days of the year?

What is the average value of $f(x) = \cos(x)$ over the interval $[0, \frac{\pi}{2}]$?

A particle moves along the x-axis with velocity given by $v(t) = t^2 - 4$. What is the total distance traveled by the particle from $t = 0$ to $t = 3$?