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IB Mathematics: Applications and Interpretation HL · 13.5: Integration · 13.5.9

Integration: exam review Exam review: second paper Answers

Section A: short questions

    1. (a) 27 m
    2. (b) 139.5 m
    1. (a) \(2x^{4} + 6x + \frac{5}{x^{3}} + c\)
    2. (b) \(x^{3} + \frac{5}{2}x^{2} - 2x + c\)
    3. (c) \(4x^{3} - 5x - \frac{9}{x} + c\)
    1. (a) \(6x^{\frac{3}{2}} + 20\sqrt{x} + c\)
    2. (b) \(y=4x^{\frac{3}{2}} - 7x - 25\)
    1. (a) \(-\dfrac{7}{3x^{3}}+c\)
    2. (b) \(x^{6}-4x^{2}+\dfrac{4}{x^{2}}\); \(\dfrac{x^{7}}{7}-\dfrac{4x^{3}}{3}-\dfrac{4}{x}+c\)
    3. (c) \(g(x)=12x^{3}+\dfrac{24}{x^{4}}\)
    1. (a) \(y=4x^{3} + 3x + \frac{3}{x^{3}} + 4\)
    2. (b) 42.4
    1. (a) \(f(x)=x^{4}-3x^{2}+4\)
    2. (b) 6.4
    1. (a) \(\displaystyle\int_{2}^{6}\left(x^{2} - 8x + 18\right)dx\)
    2. (b) 13.3
    3. (c) 18.7
    1. (a) \(r=9t\)
    2. (b) 4050 litres
    1. (a) 8.963, 14.78, 24.36
    2. (b) 35.5
    3. (c) Greater than the exact value: the curve bends upwards, so the tops of the trapeziums lie above the curve.
    1. (a) 4080 m²
    2. (b) 804 m²
    3. (c) 3280 m²
    1. (a) 8
    2. (b) 16

Section B: long questions

    1. (a) \(k=0.8\)
    2. (b) 85 litres
    3. (c) \(t=8.30\) minutes
    4. (d) The amount that flows in between \(t=3\) and \(t=7\): 22 litres
    5. (e) The rate keeps increasing for ever, which is not realistic: a pump cannot run faster and faster without limit.
    1. (a) The distance travelled
    2. (b) 556 m
    3. (c) Less than the true distance: the curve bends downwards, so the tops of the trapeziums lie below it.
    4. (d) 576 m
    5. (e) 3.52%
    6. (f) 11.2 s
    1. (a) \(x=12\)
    2. (b) \(x=8\)
    3. (c) \(\displaystyle\int_0^{8}\left(8x-x^{2}\right)dx\)
    4. (d) 85.3 m²
    5. (e) 288 m²
    6. (f) 29.6%
    7. (g) 68.3 m³
    1. (a) \(x=2.23\) and \(x=10.8\)
    2. (b) \(\displaystyle\int_{2.23}^{10.8}\left((13-x)-\frac{24}{x}\right)dx\); 17.7 m²
    3. (c) 32.0 m²
    4. (d) 31.2
    5. (e) 2.53%
    6. (f) 8.86 m³

jmaths.xyz · AI HL · 13.5.9 Unit review