The Eiffel Tower property Let R be the region between the curves y = e?cx and y = ?e?cx on the interval [a, ?), where a ? 0 and c > 0. The center of mass of R is located at (, 0) where . (The profile of the Eiffel Tower is modeled by the two exponential curves.)a. For a = 0 and c =2, sketch the curves that define R and find the center of mass of R. Indicate the location of the center of mass.________________b. With a = 0 and c = 2, find equations of the tangent lines to the curves at the points corresponding to x =0.________________c. Show that the tangent lines intersect at the center of mass.________________d. Show that this same properly holds for any a ? 0 and any c > 0; that is, the tangent lines to the curves y = ±e?cx at x = a intersect at the center of mass of R.(Source: P. Weidman and I. Pinelis, Complex Rendu, Mechanique 332 (2004): 571–584. Also see the Guided Projects.)

Problem 72EThe Eiffel Tower property Let R be the region between the curves and on the interval [a, ). where a 0 and c > 0. The center of mass of R is located at (,) where . (The profile of the Eiffel Tower is modeled by the two exponential curves.)a. For a = 0 and c =2, sketch the curves that define R and find the center of mass of R. Indicate the location of the center of mass.b. With a = 0 and c = 2, find equations of the tangent lines to the curves at the points corresponding to x =0.c. Show that the tangent lines intersect at the center of mass.d. Show that this same properly holds for any a 0 and any c > 0; that is, the tangent lines to the curves at x = a intersect at the center of mass of R.Solution:Step 1a. For a = 0 and c =2, sketch the curves that define R and find the center of mass of R. Indicate the location of the center of mass.For this part, we draw the graph and locate the centre of mass of R.To locate the center of mass, we evaluate the integrals as follows, (Using Method of Integration by Parts)Taking the value of a=0, and computing the limits, we get Taking the value of a=0, and computing the limits, we get Hence the center of mass The graph of the region R is as follows, The black dot represents the center of mass of the region R which is bounded by red, purple curve and y-axis.