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

Verified step by step guidance

1. Identify the interval over which the object extends. Typically, for a density function \( \delta(x) = \frac{4}{\sqrt{x}} \), the domain is \( x > 0 \). Confirm the specific interval \([a, b]\) for the problem, as the center of mass depends on this range.
2. Recall the formula for the center of mass \( \bar{x} \) of a one-dimensional object with variable density \( \delta(x) \): \[ \bar{x} = \frac{\int_a^b x \delta(x) \, dx}{\int_a^b \delta(x) \, dx} \] This formula represents the weighted average position, where the weights are given by the density function.
3. Set up the numerator integral: \[ \int_a^b x \cdot \frac{4}{\sqrt{x}} \, dx = \int_a^b 4x^{1 - \frac{1}{2}} \, dx = \int_a^b 4x^{\frac{1}{2}} \, dx \] This integral calculates the moment of the mass distribution about the origin.
4. Set up the denominator integral: \[ \int_a^b \frac{4}{\sqrt{x}} \, dx = \int_a^b 4x^{-\frac{1}{2}} \, dx \] This integral calculates the total mass of the object over the interval \([a, b]\).
5. Evaluate both integrals using the power rule for integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for} \quad n \neq -1 \] After evaluating, substitute the limits \(a\) and \(b\) into both integrals, then divide the numerator by the denominator to find the center of mass \( \bar{x} \).

Key Concepts

Center of Mass

The center of mass is the point at which the weighted position of a body or system balances. For a one-dimensional object with variable density, it is found by taking the ratio of the moment (integral of position times density) to the total mass (integral of density). This concept generalizes the idea of the average position weighted by mass distribution.

Density Function

A density function δ(x) describes how mass is distributed along an object as a function of position x. When density varies, it affects the calculation of total mass and moments, requiring integration of δ(x) over the object's domain. In this problem, δ(x) = 4/√x indicates density increases as x approaches zero.

Definite Integrals in Calculus

Definite integrals are used to compute total quantities like mass and moments when density varies continuously. Integrating δ(x) over an interval gives total mass, while integrating x·δ(x) gives the moment about the origin. These integrals are essential for finding the center of mass with variable density.