Question

this question has several parts that must be completed sequentially. if you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.
estimate
(int _ { 0 } ^ { 1 } 2 cos left( x ^ { 2 }
ight) d x)
using the following rule with
(n = 4).
exercise (a)
the trapezoidal rule
click here to begin!
exercise (b)
the midpoint rule
step 1
the midpoint rule says that
(int _ { a } ^ { b } f ( x ) d x = m _ { n } = delta x left f left( overline { x _ { 1 } }
ight) + f left( overline { x _ { 2 } }
ight) + ldots + f left( overline { x _ { n } }
ight)
ight)
with
(delta x = \frac { b - a } { n }).
we again need to estimate
(int _ { 0 } ^ { 1 } 2 cos left( x ^ { 2 }
ight) d x)
with
(n = 4)
subintervals.
for this,
(delta x = \frac { 1 - 0 } { square } = square).

Explanation:

Step1: Calculate the width of sub - intervals

The formula for the width of sub - intervals in numerical integration is \(\Delta x=\frac{b - a}{n}\). Here, \(a = 0\), \(b = 1\), and \(n = 4\).

$$ \Delta x=\frac{1-0}{4}=\frac{1}{4}=0.25 $$

Answer:

\(0.25\)