Applications of Integration

The rate at which a population of birds is growing is given by the function $P(t) = 2t^2 + 3t + 1$, where $t$ represents time in years. What is the total increase in the bird population from $t = 1$ to $t = 3$?

If $F(x)$ is an accumulation function given by $F(x) = \int_0^x f(t) dt$, and $f(t)$ has a discontinuity at $t=3$, what must be true about the differentiability of $F(x)$ at $x=3$?

If $\int_0^a f'(x) dx = \int_a^b f'(x) dx$, what must be true about $f(a)$?

For a certain chemical reaction, if temperature ($T$), measured in degrees Celsius as a function of time ($\tau$, measured in minutes), follows a model given by $T(\tau)=100e^{-\tau}$ starting from $\tau = 0$, how much does temperature decrease between minute one and two?

If the rate of water flow into a tank is given by the function $f(t) = 3t^2 - t^3$ liters per minute from time $t = 0$ to $t = 4$ minutes, what is the total amount of water that flows into the tank over this period?

The velocity of an object moving along a straight line is given by the function $v(t) = 2t + 1$, where $t$ represents time in seconds. What is the total displacement of the object from $t = 0$ to $t = 5$?

If $g(z)=\int_{z^2}^{\cos(\pi u)} du$ from $u=z$ to $u=-z$ exhibits odd symmetry about $z =0$, what can we deduce about $g(z)$?

What is the result of the definite integral $\int _0 ^\pi \sin(x)dx$?

A car is traveling at a constant speed of 60 miles per hour. How far does the car travel in 3 hours?

The velocity of an object is given by the function $v(t) = t^2 - t$, where $t$ represents time in seconds. What is the total displacement of the object from $t = 1$ to $t = 3$?