Question:
The area of the surface
\[
z = \frac{xy}{3}
\]
intercepted by the cylinder
\[
x^2 + y^2 \leq 16
\]
lies in the interval
\[
\left( 20\pi, 22\pi \right)
\]
The area of the surface
\[
z = \frac{xy}{3}
\]
intercepted by the cylinder
\[
x^2 + y^2 \leq 16
\]
lies in the interval
\[
\left( 20\pi, 22\pi \right)
\]
Show Hint
For surface area integrals, remember to use the correct formula and parametrize the region carefully.
Updated On: Nov 18, 2025
- \( (20\pi, 22\pi) \)
- \( (22\pi, 24\pi) \)
- \( (24\pi, 26\pi) \)
- \( (26\pi, 28\pi) \)
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The Correct Option is A
Solution and Explanation
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).
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).
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