Let be the portion of the graph of from to with .

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

Problem

Let

C

be the portion of the graph of

y=\cosh x

from

x=0

x=a

with

a>0

. (a) Compute

\dfrac{dy}{dx}

for

y=\cosh x

. Show all steps. (b) Verify algebraically that

\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\cosh x

for every real

x

. Show each identity you use. (c) Explain, without carrying out the integration, why the result in part (b) implies that the arc length of

C

can be obtained by evaluating

\displaystyle \int_0^{a} \cosh x\,dx

. Your explanation should refer explicitly to the definition of arc length for a function

y=f(x)