1. 7.4 Lengths of Curves and Surface Area
(Photo not taken by Vickie Kelly)
Greg Kelly, Hanford High School, Richland, Washington

2. 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.

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

4. 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.

5. You may want to let the calculator find the derivative too:
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

6. Example:

7. Notice that x and y are reversed.
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.

8. 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!

9. Example:
Rotate about the y-axis.

10. Example:
Rotate about the y-axis.

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

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

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

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