Find the area of the portion of the surface $f ( x , y ) = \sqrt { 12 - x ^ { 2 } - y ^ { 2 } } \text { that lies above }$ the region $R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 144 \right\}$ Round your answer to two decimal places.
A)144.00
B)118.44
C)904.78
D)452.39
E)1,421.22

Question

Quizplus · Accepted Answer

The surface described by $f(x, y) = \sqrt{12 - x^2 - y^2}$ is a hemisphere with a radius of $\sqrt{12}$. The region $R$ described by $x^2 + y^2 \leq 144$ is a circular disk with a radius of 12, which is much larger than the radius of the hemisphere. Therefore, the area of the portion of the surface lying above $R$ is actually the surface area of the entire hemisphere. The surface area of a hemisphere is given by $2\pi r^2$, where $r$ is the radius of the hemisphere. Thus, the surface area is $2\pi(\sqrt{12})^2 = 2\pi(12) = 24\pi$. The numerical value of $24\pi$ is approximately 75.398 (using $\pi \approx 3.14159$), but this calculation does not match the context of the question, indicating a misunderstanding in the approach.The correct approach involves recognizing that the question seems to ask for the surface area of a hemisphere, but the given function and the region $R$ do not match this interpretation directly due to the misinterpretation of the function's domain and the actual calculation for the surface area. The correct calculation involves integrating over the given region $R$, but the provided options and the initial calculation hint at a misunderstanding.Given the nature of the question and the options provided, it seems there was a mistake in the explanation process. The correct approach to find the area of the portion of the surface above the region $R$ would involve setting up and evaluating a surface integral over the region $R$, considering the limits of integration and the formula for the surface area of a hemisphere or using the formula for the area of a sphere's surface, $4\pi r^2$, and calculating a portion of it based on the given constraints. However, without the correct application of these principles, the initial explanation does not lead to a correct answer choice as provided.Let's correct the approach: The question likely involves a misunderstanding or misinterpretation of the given function and region $R$. The function $f(x, y) = \sqrt{12 - x^2 - y^2}$ represents a hemisphere, but the calculation of its surface area or the area of the portion above $R$ was not correctly addressed in the initial explanation. The correct method involves integrating over the appropriate bounds, which would require recognizing the nature of the surface and the region $R$. Given the options and the context, a reevaluation of the problem statement and the mathematical approach is necessary for an accurate answer.

Find the Area of the Portion of the Surface f(x,y)=12−x2−y2 that lies above f ( x , y ) = \sqrt { 12 - x ^ { 2 } - y ^ { 2 } } \text { that lies above }f(x,y)=12−x2−y2​ that lies above

Find the Area of the Portion of the Surface $f ( x , y ) = \sqrt { 12 - x ^ { 2 } - y ^ { 2 } } \text { that lies above }$