Question

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.
int_{-1}^{1}left(2 x^{2}+1
ight) d x
(round to two decimal places as needed.)
c. use the formula ( left(left|e_{t}
ight| /(\text { true value })
ight) \times 100 ) to express ( left|e_{t}
ight| ) as a percentage of the integrals true value.
( 5 % )
(round to the nearest integer as needed.)
ii. using simpsons rule
a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( left|e_{s}
ight| ).
s=\frac{10}{3}
(type an exact answer. type an integer or a simplified fraction.)
the upper bound for ( left|e_{s}
ight| ) is ( square ).

Explanation:

Step1: Find the fourth - derivative of \(f(x)=2x^{2}+1\)

First, \(f(x) = 2x^{2}+1\), \(f^{\prime}(x)=4x\), \(f^{\prime\prime}(x) = 4\), \(f^{\prime\prime\prime}(x)=0\), \(f^{(4)}(x)=0\)

Step2: Use the error formula for Simpson's rule

The error formula for Simpson's rule is \(|E_{S}|\leq\frac{(b - a)^{5}}{180n^{4}}\max|f^{(4)}(x)|\) on the interval \([a,b]\). Here \(a=-1\), \(b = 1\), \(n = 4\)

Since \(|f^{(4)}(x)|=0\)

Answer:

The upper - bound for \(|E_{S}|\) is \(0\)