Area bounded by the curves $y = \left[\frac{x^2}{64} + 2\right]$ ([$.$] denotes the greatest integer function), $y = x - 1$ and $x = 0$ above the $x$-axis is:
Step-by-Step Solution
Key Concept: Identify the piecewise-defined boundaries and their intersections to set up appropriate integrals for different regions.
The region is bounded by $y = 2$ for $-8 \leq x < 8$, and $y = x - 1$ for $x \geq 8$. Key intersections are at $(-8, 8)$ where $y = x - 1$ meets the left boundary, and at $x = 3, y = 2$ where $y = x - 1$ intersects $y = 2$. The area consists of rectangular and linear regions with total area computed by integration over the specified intervals.
Correct Answer: 3