8.2 | Applications to Geometry


Let \(f\) be a continuous function defined on \([a,b],\) and let \(P=\{x_0,x_1, \ldots, x_n \}\) be a regular partition of \([a,b].\) The arc length of the graph of \(f\) from \(P(a,f(a))\) to \(Q(b,f(b))\) is
$$
L = \lim_{n \rightarrow \infty} \sum_{k=1}^ \infty d(P_{k-1},P_k)
$$
if the limit exists.
Let \(f\) be smooth on \([a,b].\) Then the arc length of the graph of \(f\) from \(P(a,f(a))\) to \(Q(b,f(b))\) is
$$
L = \int_a^b \sqrt{1+ [f'(x)]^2} \; dx.
$$
Let \(f\) be smooth on \([a,b].\) The arc length function \(s\) for the graph of \(f\) is defined by
$$
s(x) = \int_a^x \sqrt{1 + [f'(t)]^2} \; dt
$$
with domain \([a,b].\)
The surface area of the surface of revolution obtained by revolving the graph of \(y = f(x)\) from \(x =a\) to \(x = b\) about the \(x\)-axis is given by
$$
s = \int_a^b 2 \pi f(x) \sqrt{1+[f'(x)]^2} \; dx
$$

E 8.2 Exercises