a lamina in the xy plane is bounded by y=0,x=0 and 2x + y=2 has mass density at (x,y) equal to the distance to the x-axis. find the mass of the lamina.
the part that is tripping me up is the mass density
To find the mass of the lamina, we first need to determine the area and then integrate the product of the mass density and area over the lamina.
Let's start by finding the equation of the line 2x + y = 2 in terms of y:
2x + y = 2
2x = 2 - y
x = (2 - y)/2
Now we can establish the limits of integration for x and y:
Since the lamina is bounded by y = 0, x = 0, and 2x + y = 2, we have:
0 ≤ x ≤ (2 - y)/2
0 ≤ y ≤ 2
The area element (dA) for the lamina is given by dA = dx * dy. Since the density at each point is equal to the distance to the x-axis, we can express it as ρ(x, y) = y.
Now, let's set up the integral to calculate the mass (M):
M = ∬ ρ(x, y) * dA
M = ∬ y * dx * dy
Integrating with respect to x, we have:
M = ∫[0 to 2] ∫[0 to (2-y)/2] y dx dy
Evaluating the inner integral:
M = ∫[0 to 2] [yx] from x = 0 to (2-y)/2 dy
Simplifying the expression:
M = ∫[0 to 2] [(2 - y)y/2] dy
M = 1/2 ∫[0 to 2] (2y - y²) dy
M = 1/2 [y² - y³/3] from 0 to 2
M = 1/2 [(2² - 2³/3) - (0² - 0³/3)]
M = 1/2 [(4 - 8/3) - 0]
M = 1/2 (12/3 - 8/3)
M = 4/2
M = 2
Therefore, the mass of the lamina is 2 units.
To find the mass of the lamina, we need to integrate the mass density over the entire region.
First, let's understand the concept of mass density. Mass density is defined as the mass per unit area. In this case, the mass density is equal to the distance from each point on the lamina to the x-axis. This means that the mass per unit area increases proportionally with the distance from the x-axis.
To calculate the mass of the lamina, we will integrate the mass density over the region bounded by the given curves.
The region is bounded by:
1. y = 0 (x-axis)
2. x = 0 (y-axis)
3. 2x + y = 2
Let's find the limits of integration for the x and y variables.
To find the limits for x:
From the equation 2x + y = 2, we can solve for x:
2x = 2 - y
x = (2 - y) / 2
Therefore, the limits for x will be from x = 0 (y-axis) to x = (2 - y) / 2 (equation of the line).
To find the limits for y:
Since y = 0 bounds the lamina from the bottom (x-axis), and given that the equation y = 2 - 2x represents the line which is the upper boundary of the lamina (which is y = 2 when x = 0), we can find the limits for y.
The limits for y will be from y = 0 (x-axis) to y = 2 - 2x (equation of the line).
Now, let's calculate the mass of the lamina using double integrals.
The mass (M) of the lamina is given by the double integral:
M = ∬ρ(x, y) dA,
where ρ(x, y) represents the mass density and dA represents the differential area element.
In this case, the mass density ρ(x, y) is equal to the distance to the x-axis at each point, which is y. Therefore, ρ(x, y) = y.
The differential area element dA can be expressed as dA = dx dy.
Now, we can rewrite the mass integral:
M = ∬y dA,
To evaluate this double integral, we integrate with respect to x first and then with respect to y:
M = ∫[0, 2] ∫[0, (2 - y) / 2] y dx dy.
Evaluating this integral will give us the mass of the lamina.