The required area is $A = \int_{0}^{1} x^2(z - 1)^2 dx$
Step-by-Step Solution
Key Concept: Finding the area between curves requires setting up the integral with the difference of functions and evaluating the definite integral.
The required area is calculated as $A = \int_{0}^{1} x^2(z - 1)^2 dx = \int_{0}^{1} (x^4 + z^2 - 2zx) dx$. Expanding and integrating term by term: $\left[\frac{x^5}{5} + \frac{z^2x}{1} - \frac{2zx^2}{2}\right]_0^1 = \frac{1}{5} + z^2 - z^2 = k$. Therefore $60k = 2$.
Correct Answer: 2