We are asked to analyze the iterated integral $\int_0^2 \int_0^x k \, dy \, dx$. The goal is to sketch the lamina $R$, determine the density $\delta$, and find the mass and center of mass.

1. Problem Description

We are asked to analyze the iterated integral $\int_0^2 \int_0^x k \, dy \, dx$. The goal is to sketch the lamina $R$, determine the density $\delta$, and find the mass and center of mass.

2. Solution Steps

The iterated integral $\int_0^2 \int_0^x k \, dy \, dx$ represents a double integral over a region $R$ in the $xy$-plane.

First, we need to identify the region of integration.

The limits of integration for $y$ are $0 \le y \le x$.

The limits of integration for $x$ are $0 \le x \le 2$.

This describes the region bounded by the lines $y=0$, $y=x$, and $x=2$. This is a triangle with vertices at $(0,0)$, $(2,0)$, and $(2,2)$.

The density $\delta$ is given by the integrand, which is $k$. Since $k$ is a constant, the density is constant.

The mass $M$ is given by the double integral:

$M = \int_0^2 \int_0^x k \, dy \, dx = k \int_0^2 \int_0^x dy \, dx = k \int_0^2 [y]_0^x dx = k \int_0^2 x \, dx = k \left[ \frac{1}{2}x^2 \right]_0^2 = k \left( \frac{1}{2}(2^2) - 0 \right) = 2k$.

The $x$-coordinate of the center of mass, $\bar{x}$, is given by

$\bar{x} = \frac{1}{M} \int_0^2 \int_0^x xk \, dy \, dx = \frac{1}{2k} \int_0^2 xk \int_0^x dy \, dx = \frac{1}{2} \int_0^2 x \left[ y \right]_0^x dx = \frac{1}{2} \int_0^2 x^2 \, dx = \frac{1}{2} \left[ \frac{1}{3}x^3 \right]_0^2 = \frac{1}{2} \left( \frac{8}{3} \right) = \frac{4}{3}$.

The $y$-coordinate of the center of mass, $\bar{y}$, is given by

$\bar{y} = \frac{1}{M} \int_0^2 \int_0^x yk \, dy \, dx = \frac{1}{2k} \int_0^2 k \int_0^x y \, dy \, dx = \frac{1}{2} \int_0^2 \left[ \frac{1}{2}y^2 \right]_0^x dx = \frac{1}{2} \int_0^2 \frac{1}{2}x^2 \, dx = \frac{1}{4} \int_0^2 x^2 \, dx = \frac{1}{4} \left[ \frac{1}{3}x^3 \right]_0^2 = \frac{1}{4} \left( \frac{8}{3} \right) = \frac{2}{3}$.

3. Final Answer

The lamina R is a triangle with vertices $(0,0)$, $(2,0)$, and $(2,2)$.

The density is $\delta = k$.

The mass is $M = 2k$.

The center of mass is $(\bar{x}, \bar{y}) = \left( \frac{4}{3}, \frac{2}{3} \right)$.