9. Graphical Applications of Integrals

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.

Arc length:

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

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.

Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = (1/x) over [c, c + 1]

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = c * x * √(25 - x²) over [0, 5]