Question

The area of the surface \[ z = \frac{xy}{3} \] intercepted by the cylinder \[ x^2 + y^2 \leq 16 \] lies in the interval \[ \left( 20\pi, 22\pi \right) \]

• A. \( (20\pi, 22\pi) \)
• B. \( (22\pi, 24\pi) \)
• C. \( (24\pi, 26\pi) \)
• D. \( (26\pi, 28\pi) \)

Hint:

For surface area integrals, remember to use the correct formula and parametrize the region carefully.

Answer 1

The Correct Option is A

Step 1: Set up the surface area integral.
To find the surface area, we use the formula: \[ A = \iint_S \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \, dA. \] Substitute \( z = \frac{xy}{3} \) and calculate the partial derivatives.
Step 2: Perform the integration.
After simplifying the integrals, we find the surface area lies within the interval \( (20\pi, 22\pi) \).
Step 3: Conclusion.
Thus, the correct answer is (A).