the trapezoidal rule says that\n int _ { a } ^ { b } f ( x ) d x approx t _ { n } = \frac { delta x } { 2 }…

the trapezoidal rule says that\n int _ { a } ^ { b } f ( x ) d x approx t _ { n } = \frac { delta x } { 2 } left f left( x _ { 0 } \right) + 2 f left( x _ { 1 } \right) + cdots + 2 f left( x _ { n - 1 } \right) + f left( x _ { n } \right) \right \nwe need to estimate ( int _ { 0 } ^ { 1 } 2 cos left( x ^ { 2 } \right) d x ) with ( n = 4 ) subintervals.\nwe have\n delta x = \ntherefore,\n \frac { delta x } { 2 } = \n

the trapezoidal rule says that\n int _ { a } ^ { b } f ( x ) d x approx t _ { n } = \frac { delta x } { 2 } left f left( x _ { 0 } \right) + 2 f left( x _ { 1 } \right) + cdots + 2 f left( x _ { n - 1 } \right) + f left( x _ { n } \right) \right \nwe need to estimate ( int _ { 0 } ^ { 1 } 2 cos left( x ^ { 2 } \right) d x ) with ( n = 4 ) subintervals.\nwe have\n delta x = \ntherefore,\n \frac { delta x } { 2 } = \n

Answer

Explanation:

Step1: Calculate (\Delta x)

For the integral (\int_{a}^{b}f(x)dx) with (n) sub - intervals, (\Delta x=\frac{b - a}{n}). Here, (a = 0), (b = 1), and (n = 4). [ \Delta x=\frac{1-0}{4}=\frac{1}{4}=0.25 ]

Step2: Calculate (\frac{\Delta x}{2})

Since (\Delta x = 0.25), then (\frac{\Delta x}{2}=\frac{0.25}{2}=0.125)

Answer:

(\Delta x = 0.25), (\frac{\Delta x}{2}=0.125)