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Integration Prep
  1. Find
    (accessible to students on the path to grade 3 or 4) [6 marks]
  2. Do not use a calculator to answer this question.
    Find the exact value of .
    (accessible to students on the path to grade 3 or 4 ) [5 marks]
  3. The diagram shows the graph of . Find the value of such that the shaded area equals 6.8 .

Ask by Cruz Phillips. in Switzerland
Jan 23,2025

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Answer

  1. The integral of with respect to is .
  2. The exact value of is .
  3. The value of such that the shaded area under from to equals is approximately .

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Mind Expander

To tackle the first integral, we can rewrite the expression to make things easier. This becomes . Now let’s write it in terms of powers: . Integrating gives us .
For the definite integral , we first find the antiderivative of , which is . Evaluating from to , we compute . Since , we get .
As for the value of in the equation such that the shaded area equals 6.8, it’s useful to set up the integral . This gives us the area under the curve from to . You would typically need to find where this integral equals 6.8, which may involve numerical methods or a graphing approach since there isn’t a simple antiderivative for that can be worked out directly.
So, get ready to roll up your sleeves and dive into those integrals! They’ll take some practice, but you’ll get the hang of it before you know it!

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