Chapter 7 - Applications of Integration - 7.1 Exercises - Page 443: 59

$$\eqalign{ & {\text{Trapezoidal rule: }}A \approx 966{\text{f}}{{\text{t}}^2} \cr & {\text{Simpson's Rule: }}A \approx 1004{\text{f}}{{\text{t}}^2} \cr} $$

$$\eqalign{ & {\text{From the image we have:}} \cr & f\left( {{x_0}} \right) = 0,{\text{ }}f\left( {{x_1}} \right) = 14,{\text{ }}f\left( {{x_2}} \right) = 14 \cr & f\left( {{x_3}} \right) = 12,{\text{ }}f\left( {{x_4}} \right) = 12,{\text{ }}f\left( {{x_5}} \right) = 15 \cr & f\left( {{x_6}} \right) = 20,{\text{ }}f\left( {{x_7}} \right) = 23,{\text{ }}f\left( {{x_8}} \right) = 25 \cr & f\left( {{x_9}} \right) = 26{\text{ }}f\left( {{x_n}} \right) = 0 \cr & {\text{Let }}\Delta x = 6{\text{ft}}{\text{, }}n = 10,{\text{ }}b - a = 10\left( 6 \right) = 60 \cr & \cr & {\text{*Using the trapezoidal rule}} \cr & A \approx \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{2n}}\left[ {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + \cdots 2f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr & {\text{Therefore}}{\text{,}} \cr & A \approx \frac{{60}}{{2\left( {10} \right)}}\left[ {0 + 2\left( {14} \right) + 2\left( {14} \right) + 2\left( {12} \right) + 2\left( {12} \right) + 2\left( {15} \right) + 2\left( {20} \right)} \right] \cr & + \frac{{60}}{{2\left( {10} \right)}}\left[ {2\left( {23} \right) + 2\left( {25} \right) + 2\left( {26} \right) + 0} \right] \cr & {\text{Simplifying}} \cr & A \approx 3\left[ {0 + 28 + 28 + 24 + 24 + 30 + 40 + 46 + 50 + 52 + 0} \right] \cr & A \approx 966{\text{f}}{{\text{t}}^2} \cr & \cr & {\text{*Using the Simpson's Rule }} \cr & \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{3n}}\left[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) + 2f\left( {{x_2}} \right) + 4f\left( {{x_3}} \right) + \cdots } \right. \cr & \left. { + 4f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr & A \approx \frac{{60}}{{3\left( {10} \right)}}\left[ {0 + 4\left( {14} \right) + 2\left( {14} \right) + 4\left( {12} \right) + 2\left( {12} \right) + 4\left( {15} \right) + 2\left( {20} \right)} \right] \cr & + \frac{{60}}{{3\left( {10} \right)}}\left[ {4\left( {23} \right) + 2\left( {25} \right) + 4\left( {26} \right) + 0} \right] \cr & A \approx 2\left[ {0 + 56 + 28 + 48 + 24 + 60 + 40 + 92 + 50 + 104} \right] \cr & A \approx 1004{\text{f}}{{\text{t}}^2} \cr} $$