Question

Find the total mass of the triangle bounded by y = x y=x, the x x axis, and x = 2 x=2, if the density is given by σ ( x , y ) = 3 x y + 1 σ(x,y)=3xy+1 grams per unit area.

Answer 1

To find the total mass of the triangle bounded by y = x, the x-axis, and x = 2, calculate the area of the triangle and multiply it by the density of the material.

Step-by-step explanation:

To find the total mass of the triangle bounded by y = x, the x axis, and x = 2, we need to calculate the area of the triangle and then multiply it by the density of the material.

The triangle is bounded by the lines y = x, the x-axis, and x = 2. To find the points of intersection, we can set y = x equal to 0 (x-axis) and solve for x, giving us x = 0. Then, we can set x = 2 equal to y and solve for y, giving us y = 2.

Next, we can calculate the area of the triangle using the formula for the area of a triangle: A = (base * height) / 2. In this case, the base is 2 and the height is 2, so the area is (2 * 2) / 2 = 2.

Finally, we can calculate the total mass by multiplying the area of the triangle (2) by the density of the material, which is given by σ(x,y) = 3xy + 1. Therefore, the total mass of the triangle is 2 * (3xy + 1) = 6xy + 2 grams.