A wire occupies the interval (measured in meters) along a straight axis and has…

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A wire occupies the interval

0\le x\le 5

(measured in meters) along a straight axis and has linear density

\lambda(x)

measured in kilograms per meter. Which of the following is most accurately represented by the expression

\displaystyle \int_{0}^{5} \lambda(x)\,dx

? (a) The mass of the wire if its density were constant and equal to

\lambda(5)

, i.e.

5\,\lambda(5)

(b) The mass of the wire if its density were approximated by the average of

\lambda(0)

and

\lambda(5)

, i.e.

\tfrac{5}{2}\,[\lambda(0)+\lambda(5)]

x=5

of an antiderivative

F

\lambda

, i.e.

F(5)