Using the Trapezoidal Rule with Empirical Data

A \(140\)-foot \(({\rm ft})\) tree trunk is cut into 20-ft logs. The diameter of each cross-sectional cut is measured and its area \(A\) recorded in the table below. (\(x\) is the distance in feet of the cut from the top of the trunk.)

Find the approximate volume of the tree trunk.

Solution The volume of the tree trunk is \(V=\int_{0}^{140}A ( x) \,dx\), where \(A( x)\) is the area of a slice at \(x.\) Since only eight data points are given, the function \(A( x)\) is not explicitly known. To approximate the volume, we partition the interval \([ 0,140]\) into seven subintervals, each of width \( \Delta x=\dfrac{140}{7}=20\). This partition corresponds to the given data. Using the Trapezoidal Rule, the approximate volume of the tree trunk is \[ \begin{eqnarray*} V= \int_{0}^{140}A( x) \,dx\approx \dfrac{1}{2}(20) \,[ A(0)+2\,A\,(20)&+&2\,A(40)+2\,A (60) +2\,A(80) \\[5pt] &+& 2\,A (100) +2\,A(120) +A\,(140)] \end{eqnarray*} \] \[ \begin{eqnarray*} V &\approx & 10\,[120+2(124)+2(128)+2(130)+2(132)+2(136)+2(144)+158]\\[5pt] &=& 18{,}660 \end{eqnarray*} \]

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The volume of the tree trunk is approximately \(18{,}660{\rm ft^{3}}.\)