Chapter 8.6, Problem 28E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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,

âˆ«01ex3dx

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

Formula used:

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

|E|â‰¤(bâˆ’a)312n2[maxÂ |f''(x)|], aâ‰¤xâ‰¤b

Calculation:

Let us examine the stated integral,

âˆ«01ex3dx

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

|E|â‰¤(bâˆ’a)312n2[maxÂ |f''(x)|], aâ‰¤xâ‰¤b

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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