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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using the Trapezoidal Rule and Simpson’s Rule In Exercises 11-20, approximate the value of the definite integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of n. Round your answers to three decimal places. See Examples 1 and 2.

0 4 1 x 2 + 1 3 d x , n = 8

(a)

To determine

To calculator: The value of the integral 041x2+13dx,n=8 by using the Trapezoidal Rule and to approximate the value of the definite integral for the indicated value of n. Round your answers to four decimal places.

Explanation

Given Information:

The definite integral is 041x2+13dx,n=8

Formula used:

Trapezoidal Rule:

If a function f is continuous on [a,b], then

abf(x)dx(ba2n)[f(x0)+2f(x0)+...+2f(xn1)+f(xn)]

Calculation:

Calculation by Trapezoidal Rule:

Consider the definite integral 041x2+13dx,n=8.

When n=8, the width of each subinterval is

408=48=12

And the end points of subintervals are,

x0=0x1=12x2=1x3=32

And,

x4=2x5=52x6=3x7=72

And,

x8=4

Thus, the graph of the function is,

By Trapezoidal Rule

041x2+13dx=42×8[1(0)2+13+2×1(12)2+13+2×1(<

(b)

To determine

To calculation: The value of the integral 041x2+13dx,n=8 by using the Simpson’s Rule and to approximate the value of the definite integral for the indicated value of n. Round your answers to four decimal places.

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