Quick questions on Average value and applications of integration: VCE Math Methods Unit 4

1. Compute the definite integral $\int_{a}^{b} f(x) \, dx$. 2. Divide by the interval length $b - a$.

In modelling contexts, average value is the natural "typical level" of a continuous quantity over time. The average temperature over a day, the average flow rate, the average drug concentration. All are integrals divided by interval length.

Average value of $f(x) = \sin(x)$ on $[0, \pi]$.

Water flowing into / out of a container. Rate in litres per minute integrated over time gives volume in litres.

The displacement of the particle on $[t_1, t_2]$ is:

The total distance travelled is the integral of speed (always non-negative):

1. Find the zeros of $v(t)$ on $[t_1, t_2]$. These are the times when the particle changes direction. 2.

If $v(t) = \frac{dx}{dt}$ and $x(0) = x_0$ is the initial position, then:

Rate in litres per minute integrated over time gives volume in litres.

Rate of change of concentration integrated gives concentration change.

Rate of change of population integrated gives population change.

Power (rate of energy transfer) integrated gives energy.

$\int_{0}^{5} (2t - 6) \, dt = [t^2 - 6t]_{0}^{5} = (25 - 30) - 0 = -5$.

$v(t) = 0$ at $t = 3$. Split at $t = 3$.

The average value formula is the integral divided by $b - a$. Reporting the integral itself as the "average value" earns no marks.