7.4 Lengths of Curves and Surface Area Greg Kelly, Hanford High School, Richland, Washington

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

Example: Now what? This doesn’t fit any formula, and we started with a pretty simple example! The TI-89 gets:

Example: If we check the length of a straight line: The curve should be a little longer than the straight line, so our answer seems reasonable. The TI-89 gets:

Example: You may want to let the calculator find the derivative too: STO Y ENTER ENTER Important: You must delete the variable y when you are done! F4 4 Y ENTER

Example:

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. STO X ENTER Notice that x and y are reversed.

r Surface Area: Consider a curve rotated about the x-axis: The surface area of this band is: r The radius is the y-value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes: Surface Area about x-axis (Cartesian): To rotate about the y-axis, just reverse x and y in the formula!

Example: rotated about x-axis. STO Y ENTER ENTER

Example: rotated about x-axis. STO Y ENTER Check: ENTER

p Don’t forget to clear the x and y variables when you are done! F4 4 , Y ENTER p

Example: Rotate about the y-axis.

Example: Rotate about the y-axis.

Example: Rotate about the y-axis. From geometry: