Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-sl/integration/13-5-6/problems/b/

jmaths

IB Mathematics: Applications and Interpretation SL · 13.5: Integration · 13.5.6

Name Date

Unit mixed practice: areas, rates and accumulation Problem set B

Show each limit and the integral you write. Use the calculator for definite integrals, and give 3 significant figures unless told otherwise.

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 the area enclosed between the x-axis and the graph of \(y=-x^2 + 9x - 14\).

  2. Find the area enclosed by the part of the graph of \(y=25-x^2\) for \(x>0\), the x-axis and the y-axis.

  3. The diagram shows the graphs of \(y=10x-x^2\) and \(y=4x\). They meet at the origin and at the point P.

    246810510152025xyy = 10x − x²y = 4xP
    1. (a) Find the coordinates of P.
    2. (b) Find the shaded area.

  4. The function f satisfies \(f'(x)=2x^2 + 3x\) and \(f(0)=4\). Find \(\int_0^{2} f(x)\,dx\).

  5. Given that \(\int f(x)\,dx=5\left(x^3 + \frac{2}{x}+c\right)\), find an expression for \(f(x)\).

  6. The diagram shows the graphs of \(y=\frac{7}{x}\) and \(y=6-x\). They meet at the points P and Q.

    12345671234567xyy = 7/xy = 6 − xPQ
    1. (a) Use your calculator to find the x-coordinates of P and Q.
    2. (b) Find the shaded area.

In context

  1. A pump fills a tank at a rate of \(3+kt^2\) litres per minute, where t is the time in minutes after it is switched on. After 5 minutes the rate is 8 litres per minute. How much water is pumped in the first 5 minutes?

  2. A fountain is switched on and, for the first 20 seconds, the rate at which water leaves it is directly proportional to the time it has been on. After 8 seconds the rate is 10 litres per second. How much water leaves the fountain in the first 20 seconds?

  3. A model suggests that ice melts into a dish at a rate of \(\frac{120}{t^3}\) millilitres per minute, where t is the time in minutes after the ice was put out. When \(t=2\) there are 15 millilitres of water in the dish.

    1. (a) Find the amount of water in the dish 10 minutes after the ice was put out.
    2. (b) Use technology to estimate the amount of water that will eventually be in the dish.
    3. (c) Give one criticism of this model.

  4. Grain is loaded into a silo at a rate of \(r=\frac{t^2}{60}\), where r is measured in tonnes per hour and t is the time in hours after loading starts. The graph is shown, shaded from \(t=0\) to \(t=36\).

    102030405101520t (h)rr = t²/60
    1. (a) Write down an integral that represents the area of the shaded region.
    2. (b) Find the area of the shaded region.
    3. (c) Interpret the meaning of the shaded area in the context of this question.
    4. (d) Find the number of hours it takes to load 100 tonnes of grain.

Stretch

  1. Find the area enclosed by the curve \(y=3x^2\), the y-axis and the line \(y=27\), for \(x\ge0\).

    −11234510152025xyy = 3x²

jmaths.xyz · AI SL · 13.5.6 Unit mixed practice: areas, rates and accumulation