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 $18{,}660{\rm ft^{3}}.$