Module 25 - Parametric Equations

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test

Lesson 25.3: Arc Length of Parametric Curves

The arc length of a segment of a curve was found in Module 19. This lesson will investigate finding the arc length of a parametric curve by using a function that you will define and store in the Y= editor.

Arc Length of a Parametric Curve

The length of a parametric curve between t = a and t = b is given by the definite integral

The TI-83 can be used to find the length of the parametric curve below for .

x = cos3(t)
y = sin3(t)

Enter the functions into the Y= editor.

• Set X1T = (cos(T))^3.
• Set Y1T = (sin(T))^3.

A Function to Compute Arc Length

Because you probably do not want to enter the complicated integral each time, an arc length function can be defined in a set of parametric equations and used for curves defined by X1T and Y1T.

• Set X2T = nDeriv(X1T,T,T). Press to paste X1T into the line.
• Set Y2T = nDeriv(Y1T,T,T). Press to paste Y1T into the line.

• Define X3T = T. (This procedure will work if you leave X3T blank.)
• Define

• Return to the Home screen and store 0 in A and /2 in B.

• Enter Y3T on the Home screen to compute the arc length.

The arc length of the curve x = cos3t, y = sin3t between t = 0 and t = /2 appears to be about 1.5 units.

 Remember that nDeriv( is only approximate. If the functions are simple enough you can use symbolic derivatives to possibly improve accuracy.

25.3.1 Find the length of this parametric curve for . You should only need to change the values stored in A and B and then enter Y3T again on the Home screen.

The graph of the parametric equations x = cos3t, y = sin3t will show why both arc lengths are the same.

• Unselect the second and third pairs of parametric equations in the Y= editor and graph the first parametric equation in a [0, 2 , 0.1] x [-3, 3, 1] x [-2, 2, 1] window.

The graph is called a hypocycloid. The symmetry of the graph shows why the arc lengths are the same for and for .

The Length of the Spiral of Archimedes

The Spiral of Archimedes is defined by the parametric equations x = tcos(t), y = tsin(t).

Find the length of the spiral for .

The parametric equations in X1T and Y1T should be changed to the new equations, however the derivatives and integral in the other parametric equations should remain the same.

• Define X1T = T · cos(T).
• Define Y1T = T · sin(T).

• Store 0 in A and 20 in B.
• Enter Y3T on the Home screen.

The approximate arc length is 202.095 units. This result is difficult to obtain with pencil and paper.

Illustrating the Spiral of Archimedes

• Graph the spiral in a [0, 20, 0.1] x [-30, 30, 10] x [-30, 30, 10] window.