   Chapter 10.2, Problem 41E

Chapter
Section
Textbook Problem

# Find the exact length of the curve.41. x = 1 + 3t2, y = 4 + 2t3, 0 ≤ t ≤ 1

To determine

To find: the exact length of the curve for the parametric equation x=1+3t2 and y=4+2t3 .

Explanation

Given:

The parametric equation for the variable x is as below.

x=1+3t2

The parametric equation for the variable y is as below.

y=4+2t3

Calculation:

The length of the curve is obtained by the formula.

L=αβ(dxdt)2+(dydt)2dt

Differentiate the variable x with respect to t .

x=1+3t2dxdt=6t

Differentiate the variable y with respect to t .

y=4+2t3dydt=6t2

Write the length of the curve formula as below.

L=αβ(dxdt)2+(dydt)2dt

Substitute (6t) for dxdt and (6t2) for dydt in the above equation.

L=αβ(dxdt)2+(dydt)2dt=01(6t)2+(6t2)2dt=

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