Final answer:
To find the mass and moments of inertia for the given lamina, calculate the mass and inertia for each part and sum them up.
Step-by-step explanation:
To find the mass and moments of inertia for the given lamina, we need to calculate the mass and the mass moment of inertia for each individual part of the lamina and then sum them up.
The region bounded by y = 7x² and y = 28 can be split into two parts: the upper part above the curve y = 7x² and the lower part below the curve y = 7x² until y = 28.
Let's start by finding the mass of each part:
- For the upper part, the area can be found by integrating from x = 0 to x = sqrt(28/7), which gives us the mass as ∫[0 to sqrt(28/7)] 4 dy = 4[sqrt(28/7)] = 8√2.
- For the lower part, the area can be found by integrating from x = sqrt(28/7) to x = sqrt(28/7), which gives us the mass as ∫[sqrt(28/7) to 1] 4 dy = 4[1 - sqrt(28/7)] = 4 - 8√2 + 4√2.
The total mass of the lamina is the sum of the masses of the two parts, which is 8√2 + 4 - 8√2 + 4√2 = 8 + 4√2.