A region R is enclosed by the curve $y=\dfrac{a}{x}$ , where $a$ is a positive constant, the line $y=\dfrac12$ , the $y$ -axis and the line $y=2$ . Find the exact area of R, giving your answer in a form involving a single logarithm.

Model answer

a ln 4

Also accepted: aln4, a ln4, 2a ln 2

Explanation

Integrate with respect to

y

, since the region is bounded left by the

y

-axis and right by the curve

x=\dfrac{a}{y}

: Area

=\int_{1/2}^{2}\frac{a}{y}dy=a[\ln y]_{1/2}^{2}=a(\ln 2-\ln\tfrac12)=a\ln 4

Derived from ZIMSEC Mathematics Paper 2, June 2008, Q1

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