1. Lengths of Curves
Section 7.4a

2. Length of a Smooth Curve
Note: This method only works with a function that has a continuous
first derivative – this property is called smoothness. A function with
a continuous first derivative is smooth.

3. Length of a Smooth Curve
The length of a segment approximating
each subinterval arc:
The sum that approximates the length
of the entire curve:
This sum is not truly a Riemann sum, but we can create one by
multiplying and dividing by :

4. Length of a Smooth Curve
Because the function is smooth, then by the Mean Value Theorem…
For some point in
Passing to the limit as the norms of the subdivisions go to zero gives the length of the curve as:
Note: This technique works equally well for expressions of x in terms of y…

5. Arc Length: Length of a Smooth curve
If a smooth curve begins at and ends at ,
then the length (arc length) of the curve is
if y is a smooth function
of x on
if x is a smooth function
of y on

6. Guided Practice What is the length of the sine curve on ?
Start by graphing the function and making an estimate…
The only way we can evaluate
this integral is numerically…
How close was your estimate?

7. Guided Practice Find the length of the curve between the points:
The derivative:
is not defined at x = 0
 Check the graph!
We need to change to x as a function of y, so that the tangent at
the origin will be horizontal (the derivative will be zero instead of
undefined):

8. Guided Practice (a) (b) Graph in: by (c) Length
For each of the following, (a) set up an integral for the length of the
curve, (b) graph the curve to see what it looks like, and (c) use NINT
to find the length of the curve.
(a)
(b) Graph in: by
(c) Length

9. Guided Practice (a) (b) Graph in: by (c) Length
For each of the following, (a) set up an integral for the length of the
curve, (b) graph the curve to see what it looks like, and (c) use NINT
to find the length of the curve.
(a)
(b) Graph in: by
(c) Length

10. Guided Practice (a) (b) Graph? No. (c) Length
For each of the following, (a) set up an integral for the length of the
curve, (b) graph the curve to see what it looks like, and (c) use NINT
to find the length of the curve.
(a)
(b) Graph? No.
(c) Length