Let be a twice-differentiable function defined on (with ) such that is even; that is

MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

Let

h

be a twice-differentiable function defined on

[-a,a]

(with

a>0

) such that

h

is even; that is,

h(-x)=h(x)

for all

x

in the domain. Assume

h'

is continuous on

[-a,a]

and

h'(x)\ne 0

for

x\ne 0

. (a) State the formal definitions of an even function and an odd function. Provide one example of each (other than

h

). (b) Prove that the integrand that appears in the arc-length integral for

y=h(x)

over

[-a,a]

is itself an even function. (c) Using part (b), show that the total length

L

of the graph of

y=h(x)

[-a,a]

can be written as

L=2\displaystyle\int_{0}^{a}\!\bigl[\dots\bigr]\,dx

, where the integrand is the same as in part (b). Explain each step of your reasoning. (d) Illustrate the result of part (c) by computing, without evaluating the integral, the most reduced integral form that would give the length of the graph of

y=\cos x

[-\pi,\pi]

. Identify the limits of integration clearly.

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