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
i. using the trapezoidal rule
a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( left|e_{t}
ight| ).
t=\frac{7}{2}
(type an exact answer. type an integer or a simplified fraction.)
the upper bound for ( left|e_{t}
ight| ) is
(round to two decimal places as needed.)

Explanation:

Step1: Find the second - derivative of the function

Given \(y = f(x)=2x^{2}+1\).
First - derivative: \(y^\prime=f^\prime(x) = 4x\).
Second - derivative: \(y^{\prime\prime}=f^{\prime\prime}(x)=4\).

Step2: Use the error formula for the trapezoidal rule

The error formula for the trapezoidal rule is \(|E_T|=\frac{(b - a)^{3}}{12n^{2}}M\), where \(a=-1\), \(b = 1\), \(n = 4\), and \(M\) is the upper bound of \(|f^{\prime\prime}(x)|\) on the interval \([a,b]\).
Since \(|f^{\prime\prime}(x)| = 4\) for all \(x\), then \(M = 4\).
Substitute \(a=-1\), \(b = 1\), \(n = 4\), and \(M = 4\) into the formula:

$$
LATEXBLOCK0
$$

Answer:

The upper bound for \(|E_T|\) is \(0.17\)