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IB Mathematics: Applications and Interpretation HL · 13.5: Integration · 13.5.5

Name Date

Accumulation: rate in, amount out Problem set B

Show the integral you use. Use the calculator for definite integrals and for solving, and give 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Find \(\int (3x+2)(x^2-5)\,dx\).

  2. Find \(\int s^3\left(2s+\frac{5}{s}\right)\,ds\).

  3. Find \(\int \frac{3x^7+4x}{6x^4}\,dx\).

  4. Find \(\int \frac{x^5+8}{x^3}\,dx\).

In context

  1. Water drips into a basin at a rate of \(0.9t^2\) millilitres per second, where t is the time in seconds. After 2 seconds the basin holds 8 millilitres. Find the volume of water in the basin after 5 seconds.

  2. The rate of growth of the mass m grams of a yeast culture is modelled by \(\dfrac{dm}{dt}=kt+0.2\), where t is the time in days. When \(t=4\) the rate of growth is 1.4 grams per day.

    1. (a) Find the value of k.
    2. (b) The culture has negligible mass when \(t=0\). Find its mass after 6 days.

  3. A rain gauge is filling at a rate of \(\dfrac{72}{t^2}\) millilitres per minute, where t is the time in minutes. The gauge is empty when \(t=1\). At what time does the gauge hold 48 millilitres?

Stretch

  1. A water butt is being filled at a rate of \(\dfrac{56}{t^2}\) litres per minute, where t is the time in minutes. The butt is empty when \(t=1\). Its capacity is 50 litres.

    1. (a) Find the volume in the butt when \(t=5\).
    2. (b) Use technology to show that the butt will overflow, and find the time when it does.

jmaths.xyz · AI HL · 13.5.5 Accumulation: rate in, amount out