Equal areas under the reciprocal curve
Equal areas under the reciprocal curve
IP 4
7 marks
The region under the curve \(y=\dfrac1x\) between \(x=p\) and \(x=q\), where \(0<p<q\), is denoted by \(A(p,q)\).
- Show that \(A(a,ka)\) depends only on \(k\), for any \(a>0\) and \(k>1\).[3]
- Hence explain why the areas \(A(1,2)\), \(A(2,4)\) and \(A(4,8)\) are equal, and state their sum.[2]
- Find the value of \(n\) for which \(A(3,n)=4A(3,6)\).[2]
Solution:
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Answer:(a) \(A(a,ka)=\ln k\) (b) each equals \(\ln2\); sum \(3\ln2\) (c) \(n=48\)
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