Find the area of the surface for the portion of the paraboloid $z = 64 - x ^ { 2 } - y ^ { 2 } $ in the first Octant.
A) $\frac { \pi } { 24 } ( 257 \sqrt { 257 } - 1 )$
B) $\frac { \pi } { 40 } ( 513 \sqrt { 513 } - 3 )$
C)$\frac { \pi } { 48 } ( 65 \sqrt { 65 } - 3 )$ d.
D)$\frac { \pi } { 48 } ( 515 \sqrt { 515 } - 3 )$
E)$\frac { \pi } { 40 } ( 65 \sqrt { 65 } - 1 )$

Question

Find the area of the surface for the portion of the paraboloid $z = 64 - x ^ { 2 } - y ^ { 2 } $ in the first Octant. 
A) $\frac { \pi } { 24 } ( 257 \sqrt { 257 } - 1 )$
B) $\frac { \pi } { 40 } ( 513 \sqrt { 513 } - 3 )$
C)$\frac { \pi } { 48 } ( 65 \sqrt { 65 } - 3 )$ d.
D)$\frac { \pi } { 48 } ( 515 \sqrt { 515 } - 3 )$
E)$\frac { \pi } { 40 } ( 65 \sqrt { 65 } - 1 )$

Quizplus · Accepted Answer

The surface area $S$ of a surface defined by a function $z = f(x, y)$ over a region $D$ in the $xy$-plane can be found using the formula:$$S = \int \int_D \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dA$$For the given paraboloid $z = 64 - x^2 - y^2$, the partial derivatives are:$$\frac{\partial z}{\partial x} = -2x$$$$\frac{\partial z}{\partial y} = -2y$$Thus, the integrand becomes:$$\sqrt{1 + (-2x)^2 + (-2y)^2} = \sqrt{1 + 4x^2 + 4y^2}$$The region $D$ in the first octant for this paraboloid is a circle with radius $r = \sqrt{64}$ in the $xy$-plane, but since we're considering the first octant, we're looking at a quarter of this circle. However, the limits of integration should be adjusted to cover the entire circular base of the paraboloid in the $xy$-plane, and then the symmetry of the problem can be used to find the area of the surface in the first octant.The correct calculation involves setting up and evaluating the double integral over the circular region with radius 8 (since $z = 0$ when $x^2 + y^2 = 64$), and then applying the formula correctly. The correct answer involves detailed integration and consideration of the limits for $x$ and $y$, which are determined by the circle $x^2 + y^2 = 64$.The correct answer, after performing the integration and applying the formula for the surface area correctly, is given by option A, which is $\frac { \pi } { 24 } ( 257 \sqrt { 257 } - 1 )$. This result comes from the detailed calculation of the surface area integral, taking into account the specific form of the paraboloid and the region over which it is defined.

Find the Area of the Surface for the Portion of the Paraboloid