Chapter 7 - Applications of Integration - 7.1 Exercises - Page 443: 60

$$\eqalign{ & {\text{Trapezoidal rule: }}A \approx 324.8{\text{m}}{{\text{i}}^2} \cr & {\text{Simpson's Rule: }}A \approx 323.73{\text{m}}{{\text{i}}^2} \cr} $$

$$\eqalign{ & {\text{From the image we have:}} \cr & f\left( {{x_0}} \right) = 0,{\text{ }}f\left( {{x_1}} \right) = 11,{\text{ }}f\left( {{x_2}} \right) = 13.5 \cr & f\left( {{x_3}} \right) = 14.2,{\text{ }}f\left( {{x_4}} \right) = 14,{\text{ }}f\left( {{x_5}} \right) = 14.2 \cr & f\left( {{x_6}} \right) = 15,{\text{ }}f\left( {{x_7}} \right) = 13.5,{\text{ }}f\left( {{x_n}} \right) = 0 \cr & {\text{Let }}\Delta x = 4{\text{mi}}{\text{, }}n = 8,{\text{ }}b - a = 8\left( 4 \right) = 32 \cr & \cr & {\text{*Using the trapezoidal rule}} \cr & A \approx \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{2n}}\left[ {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + \cdots 2f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr & A \approx \frac{{32}}{{2\left( 8 \right)}}\left[ {0 + 2\left( {11} \right) + 2\left( {13.5} \right) + 2\left( {14.2} \right) + 2\left( {14} \right)} \right] \cr & + \frac{{32}}{{2\left( 8 \right)}}\left[ {2\left( {15} \right) + 2\left( {13.5} \right) + 0} \right] \cr & {\text{Simplifying}} \cr & A \approx 2\left[ {0 + 22 + 27 + 28.4 + 28 + 30 + 27 + 0} \right] \cr & A \approx 324.8{\text{m}}{{\text{i}}^2} \cr & \cr & {\text{*Using the Simpson's Rule }}\cr & \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{3n}}\left[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) + 2f\left( {{x_2}} \right) + 4f\left( {{x_3}} \right) + \cdots } \right. \cr & \left. { + 4f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr & A \approx \frac{{32}}{{3\left( 8 \right)}}\left[ {0 + 4\left( {11} \right) + 2\left( {13.5} \right) + 4\left( {14.2} \right) + 2\left( {14} \right)} \right] \cr & + \frac{{32}}{{3\left( 8 \right)}}\left[ {4\left( {15} \right) + 2\left( {13.5} \right) + 0} \right] \cr & {\text{Simplifying}} \cr & A \approx \frac{4}{3}\left[ {44 + 27 + 56.8 + 28 + 60 + 27} \right] \cr & A \approx 323.73{\text{m}}{{\text{i}}^2} \cr} $$