Arc length:
Find the length of the curve y = x², 0 ≤ x ≤ √3/2.

81. Find the lengths of the following curves.

a. y = (x²/8) - ln(x), 4≤x≤8

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.

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

Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

Centroid of a region

Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.