Question

The corners of a rectangular plate have coordinates (2,0),(4,0),(4,2) and (2,2). The density of the plate is given by rho(x,y)=y. (a) The total mass of the plate is o⁴ .

Answer 1

To calculate the total mass of the rectangular plate with a nonuniform density of ρ(x,y) = y, we can consider the plate as a collection of small rectangles and integrate the mass of each small rectangle over the plate. Integrating with respect to x and y, the total mass of the plate is 8.

Step-by-step explanation:

To find the total mass of the rectangular plate, we need to calculate the mass of each small element of the plate and then integrate over the entire plate. The density of the plate is given by ρ(x,y) = y. We can consider the plate as a collection of small rectangles, each with a width dx and a height dy.

The mass of each small rectangle is given by dM = ρ(x,y) dA, where dA = dx*dy is the area of the rectangle. Substituting the density function, we have dM = y dx dy. To calculate the total mass, we integrate this expression over the region of the plate.

Integrating with respect to x from 2 to 4 and with respect to y from 0 to 2, we have:

m = ∫∫ y dx dy = ∫[0,2] ∫[2,4] y dx dy

= ∫[0,2] [yx] [2,4] dy = ∫[0,2] (4y - 2y) dy = ∫[0,2] 2y dy = [y^2] [0,2] = 2(2)^2 - 2(0)^2 = 8