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

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

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

Difficulty: first skills applying them stretch

Skills

  1. Find the area enclosed between the x-axis and the graph of \(y=-x^2 + 7x - 10\).

  2. Find the area enclosed by the part of the graph of \(y=16-x^2\) for \(x>0\), the x-axis and the y-axis.

  3. The diagram shows the graphs of \(y=6x-x^2\) and \(y=2x\). They meet at the origin and at the point P.

    −11234567−224681012xyy = 6x − x²y = 2xP
    1. (a) Find the coordinates of P.
    2. (b) Find the shaded area.

  4. The function f satisfies \(f'(x)=x^2 + 2x\) and \(f(0)=5\). Find \(\int_0^{3} f(x)\,dx\).

  5. Given that \(\int f(x)\,dx=3\left(x^4 - \frac{1}{x}+c\right)\), find an expression for \(f(x)\).

  6. The diagram shows the graphs of \(y=\frac{12}{x}\) and \(y=10-x\). They meet at the points P and Q.

    246810246810xyy = 12/xy = 10 − 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 3D printer lays down plastic at a rate of \(6+kt^2\) grams per minute, where t is the time in minutes after it starts. After 4 minutes the rate is 14 grams per minute. How much plastic is laid down in the first 4 minutes?

  2. For the first 30 seconds after the gate opens, grain leaves a hopper at a rate directly proportional to the time since the gate opened. After 5 seconds the rate is 15 kilograms per second. How much grain leaves the hopper in the first 30 seconds?

  3. A model suggests that water drips into a collecting tray at a rate of \(\frac{80}{t^3}\) millilitres per second, where t is the time in seconds after the tap was turned on. When \(t=1\) there are 6 millilitres in the tray.

    1. (a) Find the amount of water in the tray 4 seconds after the tap was turned on.
    2. (b) Use technology to estimate the amount of water that will eventually be in the tray.
    3. (c) Give one criticism of this model.

  4. Rainwater drains from a flooded cellar at a rate of \(r=\frac{t^2}{40}\), where r is measured in cubic metres per minute and t is the time in minutes after the pump is switched on. The graph is shown, shaded from \(t=0\) to \(t=45\).

    10203040501020304050t (min)rr = t²/40
    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 minutes it takes to drain 400 m³ of water.

Stretch

  1. Find the area enclosed by the curve \(y=2x^2\), the y-axis and the line \(y=8\), for \(x\ge0\).

    −1123246810xyy = 2x²

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