# Problems On Areas Set-4

## Problems On Areas

**Example – 7**

Find the area of the region bounded by \(x = \frac{1}{2},x = 2,y = 1n\,\,x\,\rm{and}\,y = {2^x}.\)

**Solution:** The region of interest is verystraightforward to plot:

The required area is

\[\begin{align}A &= \int\limits_{1/2}^2 {\left({{2^x} - 1nx} \right)} dx\\&= \left. {\left( {\frac{{{2^x}}}{{1nx}} -x1n\,x + x} \right)} \right|_{1/2}^2\\&= \left( {\frac{{4 - \sqrt 2}}{{1n\,2}} - \frac{5}{2}1n\,2 + \frac{3}{2}} \right)sq.\,\rm{units}\end{align}\]

**Example – 8**

Let \(f\left( x \right)\, =\,\max \left\{ {{x^2},{{\left( {1 - x} \right)}^2},\,2x\,\left( {1 - x}\right)} \right\}.\) Determine the area of the region bounded by the curve \(y = f\left( x \right),\,x - axis\,,\,x =0\,\rm{and}\,x = 1\,.\)

**Solution:** The technique to plot the curve for \(f\left( x \right)\) has been outlined in the unit on Functions. We plot all the three curves \({x^2}\), \({\left( {1 - x}\right)^2}\,\,\,\rm{and}\,\,\,\,\,2x\left( {1 - x} \right)\) on the same axes, scan the \(x-\)axis from left to right and at every point, pick out thatgraph which lies uppermost of all the three graphs. In the figure below, theheavyset curve is the curve for \(f\left( x\right):\)

We can evaluate the required area, as is clear from thefigure above, by dividing the integration interval [0, 1] into three sub-intervals:

\[\begin{align}A &= \int\limits_0^{1/3} {{{\left({1 - x} \right)}^2}} dx + \int\limits_{1/3}^{2/3} {2x\left( {1 - x} \right)dx +\int\limits_{2/3}^1 {{x^2}} } \\\\&= \frac{{19}}{{18}} +\frac{{13}}{{81}} + \frac{{19}}{{81}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(\rm{verify} \right)\\\\&= \frac{{51}}{{81}}\\\\&= \frac{{17}}{{27}}\end{align}\]

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