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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use Simpson’s Rule with n = 6 to estimate the length of the curve x = tet, y = t + et, −6 ≤ t ≤ 6.

To determine

To estimate: The length of the curve using Simpson’s Rule for the parametric equations of x=tet and x=t+et.

Explanation

Given:

The parametric equation for the variable x is as below.

x=tet

The parametric equation for the variable y is as below.

y=t+et

The range of t varies from 6 to 6.

Calculation:

Differentiate the parametric equation x with respect to t.

x=tetdxdt=(1et)

Differentiate the parametric equation y with respect to t.

y=t+etdydt=(1+et)

The length of the curve is

L=66(dxdt)2+(dydt)2dt

Substitute (1+et) for dydt and (1et) for dxdt in the above equation.

L=66(1et)2+(1+et)2dt=661e2t2et+1+e2t+2etdt=662+2e2tdt

Find the value of Δt using the formula below.

Δt=ban

Substitute the value of 6 for b, -6 for a, and 6 for n in the above equation.

Δt=6+(6)6=2

The subintervals are (6,4),(4,2),(2,0), and (0,2),(2,4),(4,6).

Use the rule of Simpson to determine length

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