# Use Simpson’s Rule with n = 6 to estimate the length of the arc of the curve with equations x = t 2 , y = t 3 , z = t 4 , 0 ≤ t ≤ 3.

### Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550
Chapter 13, Problem 7RE
Textbook Problem
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## Use Simpson’s Rule with n = 6 to estimate the length of the arc of the curve with equations x = t2, y = t3, z = t4, 0 ≤ t ≤ 3.

To determine

To estimate: The length of the arc of the curve with equations x=t2,y=t3,z=t4,0t3 .

### Explanation of Solution

Formula used:

Write the expression to find the length of the curve r(t),t1tt2 .

L=t1t2|r(t)|dt (1)

Write the required differentiation formula to find the tangent vector r(t) .

ddttn=ntn1

Write the Simpson’s rule to find the length of the curve with the function f(t) and step length Δt .

t1t2f(t)dt=Δt3[f(0)+4f(Δt)+2f(2Δt)+4f(3Δt)+...+f(nΔt)] (2)

Here,

Δt is the step length and

n is the number, which considered based on the limits of scalar parameter t .

Write the expression to find the step length.

Δt=t2t1n (3)

Write the equation of the curve as follows.

x=t2,y=t3,z=t4,0t3

Write the vector equation from the equation of the curve as follows.

r(t)=t2,t3,t4

Calculation of derivative of the vector function r(t) [r(t)] :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=t2,t3,t4 as follows.

ddt[r(t)]=ddt(t2),ddt(t3),ddt(t4)

Rewrite and compute the expression as follows.

r(t)=2t21,3t31,4t41=2t,3t2,4t3

Find |r(t)| from the vector r(t)=2t,3t2,4t3 as follows.

|r(t)|=|2t,3t2,4t3|=(2t)2+(3t2)2+(4t3)2=4t2+9t4+16t6

Calculation of length of the curve (L) :

Substitute 4t2+9t4+16t6 for |r(t)| , 0 for t1 , and 3 for t2 in equation (1),

L=03(4t2+9t4+16t6)dt

Simplify and compute the expression by using Simpson’s rule as follows.

Consider |r(t)|=4t2+9t4+16t6 as function f(t) .

f(t)=4t2+9t4+16t6 (4)

As the limits of the scalar parameter is 0 to 3, consider the number n as 6 for simple and accurate calculations.

Calculation of step length Δt :

Substitute 0 for t1 , 3 for t2 , and 6 for n in equation (3),

Δt=306=12

Rewrite the Simpson’s rule by substituting L for t1t2f(t)dt , 12 for Δt , and 6 for n in equation (2) as follows

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