8.7 Lengths of Curves Greg Kelly, Hanford High School, Richland, Washington

If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the pythagorean theorem: We need to get dx out from under the radical. Length of Curve (Cartesian)Lengths of Curves:

Example: Yucko! Now what? This doesn’t fit any formula, and we started with a pretty simple example! The TI-Nspire CAS gets:

Example: The curve should be a little longer than the straight line, so our answer seems reasonable. If we check the length of a straight line:

Example: You may want to let the TI-Nspire CAS find the derivative too: 1 st define y = -x in one of 3 ways. Pick your favorite. i) Define y = -x2 -x2 + 9 ( b, 1:Actions, 1:Define) or ii) y:= - x2 x2 + 9 (: (: is middle of right column of keys; = is on the top left) or iii) –x /h y Then press /r choose 0e3e /q1+ /r choose XeYeqeeX /·

Example: Before you do the next problem on your Nspire you should go to Tools (/c), 4:Insert, 1:Problem or to clear the variable in a-z you could press b, 1:Actions, 4:Clear a-z

If you have an equation that is easier to solve for x than for y, the length of the curve can be found the same way. Notice that x and y are reversed. Try this on your Nspire