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Applications of Integration

Secondary 4 Additional Math practice

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Applications of Integration practice questions

Secondary 4 topics from the Additional Mathematics 4049 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

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What this topic covers

Worked examples

Example 1

Evaluate ∫02(3x2+4x) dx\displaystyle\int_{0}^{2} (3x^{2} + 4x) \, dx.
Show the worked solution
  • ∫02(3x2+4x) dx=[33x3+42x2]02\int_{0}^{2} (3x^{2} + 4x) \, dx = \left[\frac{3}{3}x^{3} + \frac{4}{2}x^{2}\right]_{0}^{2}
  • =[x3+2x2]02= \left[x^{3} + 2x^{2}\right]_{0}^{2}
  • =(2)3+2(2)2= (2)^{3} + 2(2)^{2}
  • =8+8= 8 + 8
  • =16= 16

Answer: 1616

Example 2

Given that ∫07f(x) dx=6\displaystyle\int_{0}^{7} f(x) \, dx = 6, evaluate ∫07[4f(x)+4] dx\displaystyle\int_{0}^{7} [4f(x) + 4] \, dx.
Show the worked solution
  • ∫07[4f(x)+4] dx=4∫07f(x) dx+∫074 dx\int_{0}^{7} [4f(x) + 4] \, dx = 4\int_{0}^{7} f(x) \, dx + \int_{0}^{7} 4 \, dx
  • =4×6+4(7−0)= 4 \times 6 + 4(7 - 0)
  • =24+28= 24 + 28
  • =52= 52

Answer: 5252

Example 3

Find the area bounded by y=x(x+4)y = x(x + 4), the xx-axis, x=−4x = -4 and x=0x = 0.
xyy = x(x + 4)Not drawn to scale
Show the worked solution
  • The curve lies below the x-axis between x=−4 and x=0\text{The curve lies below the } x\text{-axis between } x = -4 \text{ and } x = 0
  • Area=∣∫−40x(x+4) dx∣\text{Area} = \left|\int_{-4}^{0} x(x + 4) \, dx\right|
  • =∣∫−40(x2+4x) dx∣= \left|\int_{-4}^{0} (x^2 + 4x) \, dx\right|
  • =∣[x33+2x2]−40∣= \left|\left[\frac{x^3}{3} + 2x^2\right]_{-4}^{0}\right|
  • =∣(0)−(323)∣= \left|\left(0\right) - \left(\frac{32}{3}\right)\right|
  • =323 units2= \frac{32}{3} \text{ units}^2

Answer: 32/332/3

Questions parents ask

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Which syllabus does this follow?

Additional Mathematics (4049), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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