Accumulation: rate in, amount out Problem set A

Find \(\int (4x+3)(x^2-2)\,dx\).

Find \(\int t^2\left(3t-\frac{4}{t}\right)\,dt\).

Find \(\int \frac{2x^5-3x}{4x^3}\,dx\).

Find \(\int \frac{x^4-6}{x^2}\,dx\).

Rainwater flows into a barrel at a rate of \(0.6t^2\) litres per minute, where t is the time in minutes. After 1 minute the barrel holds 12 litres. Find the volume of water in the barrel after 4 minutes.

The rate of growth of the mass m grams of a patch of algae is modelled by \(\dfrac{dm}{dt}=kt+0.4\), where t is the time in weeks. When \(t=3\) the rate of growth is 1.6 grams per week.

1. (a) Find the value of k.
2. (b) The patch has negligible mass when \(t=0\). Find its mass after 5 weeks.

A fuel tank is filling at a rate of \(\dfrac{80}{t^2}\) litres per minute, where t is the time in minutes. The tank is empty when \(t=1\). At what time does the tank hold 60 litres?

A pond is being filled at a rate of \(\dfrac{70}{t^2}\) litres per minute, where t is the time in minutes. The pond is empty when \(t=1\). Its capacity is 65 litres.

1. (a) Find the volume in the pond when \(t=10\).
2. (b) Use technology to show that the pond will overflow, and find the time when it does.