Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. ∫(1/eˣ) dx = ln eˣ + C.

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")

d. Prove this property holds for any a ≥ 0 and c > 0:

The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass

(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

69. Let f(x) = sin(eˣ).

d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

47. ∫(1 to e) (1/x) dx; n = 50

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps

d. Evaluate the remaining integral using the change of variables z = sqrt(x)

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.