Question

The area of the region bounded by the curve \(y = x^3\) and the lines \(y = 8\) and \(x = 0\) is ... square units.

• A. \(8\)
• B. \(12\)
• C. \(16\)
• D. \(10\)

Hint:

Integrate with respect to \(y\): \(x = y^{1/3}\) from \(0\) to \(8\).

Answer 1

The Correct Option is B

Step 1: Plan:
Subtract the area under the curve from the enclosing rectangle.

Step 2: Steps:
Rectangle with corners at $(0,0)$, $(2,0)$, $(2,8)$, $(0,8)$ has area $16$.
Area under $y = x^3$ from $0$ to $2$ is $\frac{x^4}{4}\Big|_0^2 = 4$.
The wanted region is the rest: $16 - 4 = 12$ square units.

Final Answer:
The area is $12$ square units, option (B). \[ \boxed{12} \]