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

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

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

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 $t = 3$.

If $v$ is negative on part of the interval, the integral is negative; that is fine for displacement but you must take absolute values for distance.

Application questions are scored partly on units. Reporting "5" when the answer is "5 litres per minute" loses contextual marks.

Paper 2 expects you to set up the integral by hand (state the integrand and the limits) and use the calculator only for evaluation. A blank "by calculator, the answer is 9.20" without the set-up usually loses set-up marks.

Find the average value of $f(x) = 4x$ on $[0, 5]$. [2 marks]

Water flows in at $r(t) = 2t + 1$ L/min for $0 \le t \le 6$. Find the total volume added. [3 marks]

A particle has velocity $v(t) = 2t - 8$ m/s for $0 \le t \le 6$ s. Find (a) the displacement and (b) the distance travelled. [2+3 marks]