Chapter 6.3, Problem 3CP

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Checkpoint 3Use the Trapezoidal Rule to estimate the value of ∫ 0 1 1 + x 2 d x such that the error in the approximation of the integral is less than 0.01.

To determine

To calculate: The approximate value of the integral 011+x2dx by Trapezoidal Rule, such that the error in the approximation is less than 0.01.

Explanation

Given Information:

The definite integral is 011+x2dx.

Formula used:

The simple power rule of differentiation,

ddx(xn)=nxn1

The chain rule of differentiation for differential function f(g(x)).

ddx(f(g(x)))=(f(g(x)))(ddx(g(x)))

The quotient rule of differentiation,

ddx(u(x)v(x))=v(x)du(x)dxu(x)dv(x)dx[v(x)]2

The approximate value of integral abf(x)dx for n equal subdivision of closed interval [a,b] by Trapezoidal Rule is,

abf(x)dx(ba2n)[f(x0)+2f(x1)+2f(x2)+2f(x3)++f(xn)]

The approximate error in Trapezoidal Rule for integral abf(x)dx is,

|E|(ba)312n2[max|fn(x)|], axb

The steps to choose the value of n subdivision to calculate error:

Step:1. Find second derivative of function f(x).

Step:2. Find the maximum of |f(x)| on interval [a,b].

Step:3. The inequality for the error as,

|E|(ba)312n2[max|fn(x)|]

Step:4. Solve for n in the inequality, for an error less than ε,

(ba)312n2[max|f(x)|]<ε

Step:5. Divide the closed interval [a,b] into n subintervals and then apply the Trapezoidal Rule.

Calculation:

Consider the integral, 011+x2dx.

Use the simple power rule and chain rule of differentiation,

ddx(f(x))=ddx(1+x2)f(x)=ddx((1+x2)12)=12(1+x2)121(ddx(2x))=x1+x2

Use the quotient rule of differentiation to differentiate the first derivative, f(x)=x1+x2

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