137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.

82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is

L = ∫(from 1 to 2)√(1 + 1/x²)dx.

84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).

In Exercises 139–142, find the length of each curve.

139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

In Exercises 139–142, find the length of each curve.

141. y = ln(cos(x)) from x = 0 to x = π/4.

147. Find the area of the region between the curve y = 2x / (1 + x²) and the interval −2 ≤ x ≤ 2 of the x-axis.

Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.