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.

Verified step by step guidance

1. Identify the region bounded by the curve \(y = \frac{1}{2\sqrt{x}}\) and the x-axis from \(x = \frac{1}{4}\) to \(x = 4\). This region lies above the x-axis and below the curve within these limits.
2. Recall that the centroid \((\bar{x}, \bar{y})\) of a planar region bounded by a curve and the x-axis can be found using the formulas: \(\displaystyle \bar{x} = \frac{1}{A} \int_a^b x f(x) \, dx\) and \(\displaystyle \bar{y} = \frac{1}{2A} \int_a^b [f(x)]^2 \, dx\), where \(A\) is the area of the region, \(f(x)\) is the function defining the upper boundary, and \([a,b]\) is the interval.
3. Calculate the area \(A\) of the region using the integral: \(\displaystyle A = \int_{1/4}^4 \frac{1}{2\sqrt{x}} \, dx\). This integral will give the total area under the curve from \(x=\frac{1}{4}\) to \(x=4\).
4. Compute the \(x\)-coordinate of the centroid using: \(\displaystyle \bar{x} = \frac{1}{A} \int_{1/4}^4 x \cdot \frac{1}{2\sqrt{x}} \, dx = \frac{1}{A} \int_{1/4}^4 \frac{x}{2\sqrt{x}} \, dx\). Simplify the integrand before integrating.
5. Compute the \(y\)-coordinate of the centroid using: \(\displaystyle \bar{y} = \frac{1}{2A} \int_{1/4}^4 \left( \frac{1}{2\sqrt{x}} \right)^2 \, dx = \frac{1}{2A} \int_{1/4}^4 \frac{1}{4x} \, dx\). Evaluate this integral to find \(\bar{y}\).

Key Concepts

Solid of Revolution

A solid of revolution is formed when a plane region is rotated about a line (axis), creating a 3D object. In this problem, the region between the curve and the x-axis is revolved around the x-axis, which helps in visualizing the volume and centroid of the solid generated.

Centroid of a Plane Region

The centroid is the geometric center or 'balance point' of a plane region. It can be found using the coordinates (x̄, ȳ), where x̄ and ȳ are calculated by integrating moments of the area about the y-axis and x-axis, respectively, divided by the total area.

Definite Integration for Area and Moments

Definite integrals are used to calculate the area under curves and the moments needed to find the centroid. For this problem, integrating the function y=1/(2√x) from x=1/4 to x=4 gives the area, and integrating x·y and y² terms helps find the moments about the axes.