Question: Consider the graph of the vector-valued function r ( t ) = t i +16 t 2 j + t 3 k on the interval

Consider the graph of the vector-valued functionr(t) =ti+16t²j+t³kon the interval [0,4]

(a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints. (Round your answer to three decimal places.)

(b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the vectors andr(0),r(1),r(2),r(3), andr(4). (Round your answer to four decimal places.)

(c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b).Increase the decimal precision of rounded answers.

Round all numbers to integers.

Decrease the number of line segments.

Increase the number of line segments.

Change the vector to a unit vector.

(d) Use the integration capabilities of a graphing utility to approximate the length of the curve. (Round your answer to five decimal places.)

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