Question

Alamina occupies the part of the rectangle 0<=x<=2,0<=y<=5 and the density at each point is given by the function rho(x,y)=8x+2y+5

What is the totai mass?

Answer 1

The total mass of the rectangle with dimensions along the x-axis from 0 to 2 and the y-axis from 0 to 5, with a density function ρ(x,y) = 8x + 2y + 5, is found to be 205 units of mass after performing the necessary double integration over the area.

Step-by-step explanation:

The question asks to find the total mass of a rectangle with variable density ρ(x, y) over the given region. The rectangle's dimensions are along the x-axis (0 ≤ x ≤ 2) and the y-axis (0 ≤ y ≤ 5). The density function is ρ(x, y) = 8x + 2y + 5. To find the total mass, we need to integrate this density function over the area of the rectangle:

Mass = ∫∫_R ρ(x, y) dA

Where R is the region of the rectangle in the xy-plane. Applying the double integral over the provided limits:

Mass = ∫²_0 ∫⁵_0 (8x + 2y + 5) dy dx

Firstly, we integrate with respect to y:

Mass = ∫²_0 [(8x)y + (y²) + 5y] ⁵_0 dx
Mass = ∫²_0 [40x + 62.5] dx

Then, integrate with respect to x:

Mass = [20x² + 62.5x] ²_0
Mass = [20(2)² + 62.5(2)] - [20(0)² + 62.5(0)]
Mass = 80 + 125
Mass = 205 (units of mass)

The total mass of the rectangular lamina is 205 units.