   Chapter 10, Problem 7PS

Chapter
Section
Textbook Problem

Cornu Spiral Consider the cornu spiral given by x ( t ) = ∫ 0 t cos π u 2 2 d u and y ( t ) = ∫ 0 t sin π u 2 2 d u .(a) Use a graphing utility to graph the spiral over the interval − π ≤ t ≤ π .(b) Show that the cornu spiral is symmetric with respect to the origin.(c) Find the length of the comu spiral from t = 0 to t = a . What is the length of the spiral from t = − π to t = π ?

(a)

To determine

T graph: The provided parametric functions x(t)=0tcosπu22du and y(t)=0tsinπu22du over the interval πtπ.

Explanation

Given:

The functions are:

x(t)=0tcosπu22duy(t)=0tsinπu22du

Graph:

Use Ti83 to draw the graph as below:

Step1: Press MODE and select PAR for parametric equation.

Step2: Press “Y=”.

Step3: Press 2nd then MATH and enter 9 to select finite integral.

Step4: enter the function in X1 as “ (fnint(cos(πT2/2)),T,0,T)

and Y1 as (fnint(sin(πT2/2)),T,0,T)

Step5: Press WINDOW and enter the values:

(b)

To determine

To prove: The Cornu spiral formed by the function x(t)=0tcosπu22du and y(t)=0tsinπu22du where πtπ is symmetric with respect to origin.

(c)

To determine

To calculate: The length of the Cornu spiral from t=0 to t=a. Also, find the length of the spiral from t=π to t=π.

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