Final answer:
To find the mass of the lamina bounded by the given functions and a uniform density, a double integral of the density function over the area delimited by these functions is performed. The closest answer to the calculated value is 2, assuming uniform density is '1'. If the density function were given and differed from '1', the answer could vary.
Step-by-step explanation:
The question asks for the mass of a lamina with boundaries given by mathematical functions and a uniform density. To find the mass, we integrate the density function over the area of the lamina. Since no specific density function is provided in this question, we'll assume a uniform density of '1'. The integration limits are from x = 0 to x = 1 for the x-axis (as per the boundary x = 1) and from y = 0 to y = 4√x (derived from the boundary equation) for the y-axis. The mass is then the double integral of the density function over these limits.
The integration step by step:
- Set up the double integral: ∫∫ dA.
- Integrate with respect to y first from 0 to 4√x.
- Integrate with respect to x from 0 to 1.
- After computing the integral, the answer is the mass of the lamina.
Performing the integral, you get:
∫ (from 0 to 1) ∫ (from 0 to 4√x) dy dx = ∫ (from 0 to 1) [4√x] dx = ∫ (from 0 to 1) 4x^0.5 dx
This gives us 4 * (2/3)x^1.5 evaluated from 0 to 1 which is 4 * (2/3)(1)^1.5 - 4 * (2/3)(0)^1.5, resulting in (8/3) or approximately 2.67. But since none of the given options perfectly match this value and considering it could be a typographical error, the closest option to this value is (d) 2, which suggests the uniform density might not be '1' as assumed or that the question may contain additional context not provided.