A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. It can be shown that the sine curve has equation $y=\sin(\pi x/7).$

Find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. Use your calculator or a CAS to evaluate the integral correct to four significant digits.

$w =$ inches