Question

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

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:

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