Barrons AP Calculus - David Bock

Shell.

(Note that we obtained the same result using washers in Example 8.)
FIGURE N7–20
NOTE: In Examples 32 and 33 we consider finding the volumes of solids using shells that lead to
improper integrals.

BC ONLY

C. ARC LENGTH

If the derivative of a function y = f (x) is continuous on the interval a x b, then the length s of the
arc of the curve of y = f (x) from the point where x = a to the point where x = b is given by

Here a small piece of the curve is equal approximately to
As Δx → 0, the sum of these pieces approaches the definite integral above.
If the derivative of the function x = g(y) is continuous on the interval c ≤ y ≤ d, then the length s of
the arc from y = c to y = d is given by

If a curve is defined parametrically by the equations x = x(t) and y = y(t), if the derivatives of the
functions x(t) and y(t) are continuous on |ta, tb], (and if the curve does not intersect itself), then the
length of the arc from t = ta to t = tb is given by