Mass and center of mass of a lamina practice problems
Mass and center of mass of a thin plate with variable density, by double integrals. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
A lamina occupies the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (3, 0) \), \( \displaystyle (0, 3) \) with density \( \displaystyle \rho(x, y) = x^{2} + y^{2} \). Find its mass and center of mass.
A lamina occupies the region between \( \displaystyle y = x^2 \) and \( \displaystyle y = x \) with density \( \displaystyle \rho(x, y) = x^{2} + y^{2} \). Find its mass and center of mass.
A lamina occupies the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (1, 0) \), \( \displaystyle (0, 1) \) with density \( \displaystyle \rho(x, y) = x^{2} + y^{2} \). Find its mass and center of mass.
A lamina occupies the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle 0 \le y \le 3 \) with density \( \displaystyle \rho(x, y) = x y \). Find its mass and center of mass.
A lamina occupies the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (2, 0) \), \( \displaystyle (0, 2) \) with density \( \displaystyle \rho(x, y) = x^{2} + y^{2} \). Find its mass and center of mass.
A lamina occupies the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle 0 \le y \le 2 \) with density \( \displaystyle \rho(x, y) = y \). Find its mass and center of mass.
A lamina occupies the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle 0 \le y \le 1 \) with density \( \displaystyle \rho(x, y) = x^{2} + y^{2} \). Find its mass and center of mass.
A lamina occupies the region between \( \displaystyle y = x^2 \) and \( \displaystyle y = x \) with density \( \displaystyle \rho(x, y) = x + y \). Find its mass and center of mass.
A lamina occupies the rectangle \( \displaystyle 0 \le x \le 2 \), \( \displaystyle 0 \le y \le 1 \) with density \( \displaystyle \rho(x, y) = x + 1 \). Find its mass and center of mass.
A lamina occupies the region between \( \displaystyle y = x^2 \) and \( \displaystyle y = x \) with density \( \displaystyle \rho(x, y) = x y \). Find its mass and center of mass.