Chapter 4 - Integration - 4.6 Exercises - Page 310: 17

$$\eqalign{ & {\text{Trapezoidal Rule}} \approx - 0.09754 \cr & {\text{Simpson's Rule}} \approx - 0.09773 \cr & {\text{Graphing utility}} \approx - 0.097731 \cr} $$

$$\eqalign{ & \int_3^{3.1} {\cos {x^2}} dx \cr & {\text{*Using the trapezoidal Rule }}\left( {{\text{THEOREM 4}}{\text{.17}}} \right) \cr & \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{For }}n = 4,{\text{ }}\Delta x = \frac{{b - a}}{n} = \frac{{3.1 - 3}}{4} = \frac{{0.1}}{4},{\text{ then,}} \cr & {x_0} = 3,{\text{ }}{x_1} = 3.025,{\text{ }}{x_2}{\text{ = }}3.05{\text{, }}{x_3} = 3.075,{\text{ }}{x_4} = 3.1 \cr & f\left( {{x_0}} \right) = f\left( 3 \right) = \cos {\left( 3 \right)^2} \cr & f\left( {{x_1}} \right) = f\left( {3.025} \right) = \cos {\left( {3.025} \right)^2} \cr & f\left( {{x_2}} \right) = f\left( {3.05} \right) = \cos {\left( {3.05} \right)^2} \cr & f\left( {{x_3}} \right) = f\left( {3.075} \right) = \cos {\left( {3.075} \right)^2} \cr & f\left( {{x_4}} \right) = f\left( {3.1} \right) = \cos {\left( {3.1} \right)^2} \cr & {\text{Therefore,}} \cr & \int_3^{3.1} {\cos {x^2}} dx \approx \frac{{0.1}}{{2\left( 4 \right)}}\left[ {\cos {{\left( 3 \right)}^2} + 2\cos {{\left( {3.025} \right)}^2} + 2\cos {{\left( {3.05} \right)}^2}} \right] \cr & + \frac{{0.1}}{{2\left( 4 \right)}}\left[ {2\cos {{\left( {3.075} \right)}^2} + \cos {{\left( {3.1} \right)}^2}} \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_3^{3.1} {\cos {x^2}} dx \approx - 0.09754 \cr & \cr & {\text{*Using the Simpson's Rule }}\left( {{\text{THEOREM 4}}{\text{.19}}} \right) \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 & \int_3^{3.1} {\cos {x^2}} dx \approx \frac{{0.1}}{{3\left( 4 \right)}}\left[ {\cos {{\left( 3 \right)}^2} + 4\cos {{\left( {3.025} \right)}^2} + 2\cos {{\left( {3.05} \right)}^2}} \right] \cr & + \frac{{0.1}}{{3\left( 4 \right)}}\left[ {4\cos {{\left( {3.075} \right)}^2} + \cos {{\left( {3.1} \right)}^2}} \right] \cr & {\text{Simplifying by using a calculator}} \cr & \int_3^{3.1} {\cos {x^2}} dx \approx - 0.09773 \cr & \cr & {\text{Using a graphing utility we obtain}} \cr & \int_3^{3.1} {\cos {x^2}} dx \approx - 0.097731 \cr} $$