   Chapter 12.5, Problem 15E

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# Investigation Consider the graph of the vector-valued function r ( t ) = t i + ( 4 − t 2 ) j + t 3 k on the interval [0, 2].(a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints.(b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the vectors r ( 0 ) , r ( 0.5 ) , r ( 1 ) , r ( 1.5 ) ,   and   r ( 2 ) .(c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b).(d) Use the integration capabilities of a graphing utility to approximate the length of the curve. Compare this result with the answers in parts (a) and (b).

(a)

To determine

To Calculate: The length of the line segment connecting its endpoints.

Explanation

Given:

The provided function and the interval are, r(t)=ti+(4t2)j+t3k [0, 2] respectively.

Formula Used:

Length of the curve s -as::

s=(x2x1)2+(y2y1)2+(z2z1)2

Calculation:

The provided vector-function for the path is,

r(t)=ti+(4t2)j+t3k

The coordinates for the end point of the provided interval are:

r(0)=</

(b)

To determine

To Calculate: The length of the curve by summing up the lengths of the line segments connecting the terminal points of the vectors r(0),r(0.5),r(1),r(1.5) and r(2).

(c)

To determine
A more accurate approximation for (a) and (b).

(d)

To determine

To Calculate: The value of 02r(t)dt

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