   Chapter 8.6, Problem 28E

Chapter
Section
Textbook Problem

Estimating Errors In Exercises 25-28, use the error formulas in Theorem 8.6 to estimate the errors in approximating the integral, with n = 4 , using (a) the Trapezoidal Rule and (b) Simpson’s Rule. ∫ 0 1 e x 3 d x

(a)

To determine

To calculate: Errors in approximating the integral 01ex3dx while applying Trapezoidal Rule for n=4.

Explanation

Given:

The stated integral is,

01ex3dx

The value of n for estimating the error is n=4

Formula used:

The error E in approximating abf(x)dx by the Trapezoidal Rule is specified by,

|E|(ba)312n2[max |f''(x)|], axb

Calculation:

Let us examine the stated integral,

01ex3dx

Now, if f has a constant second derivative on [a,b], then the error E in approximating abf(x)dx by the Trapezoidal Rule is specified by,

|E|(ba)312n2[max |f''(x)|], axb

So, start by letting f(x)=ex3 and finding the second derivative of f

Thus,

f'(x)=3x2ex3f''(x)=3(2x+3x4)ex3

Thus, f''(x) is constant on the interval [0,1] as it is definite for all values of x in the interval [0,1] so there are no break points

(b)

To determine

To calculate: Errors in approximating the integral 01ex3dx using Simpson’s Rule for n=4.

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