The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 25$ by x = 0, x = 1, z = 1, and z = 3 is

A) $10 \sin ^ { - 1 } \left( \frac { 1 } { 5 } \right)$
B) $10 \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)$
C) $10 \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$
D) $10 \sin ^ { - 1 } \left( \frac { 4 } { 5 } \right)$
E) $10 \sin ^ { - 1 } \left( \frac { 6 } { 5 } \right)$

Question

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 25$ by x = 0, x = 1, z = 1, and z = 3 is

A) $10 \sin ^ { - 1 } \left( \frac { 1 } { 5 } \right)$
B) $10 \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)$
C) $10 \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$
D) $10 \sin ^ { - 1 } \left( \frac { 4 } { 5 } \right)$
E) $10 \sin ^ { - 1 } \left( \frac { 6 } { 5 } \right)$

Quizplus · Accepted Answer

The surface area is given by the integral of the square root of 1 plus the square of the partial derivatives of the function with respect to x and y. The partial derivatives are $2x$ and $2y$, so the surface area is$$ \int_0^1 \int_1^3 \sqrt{1 + 4x^2 + 4y^2} \ dy \ dx $$This integral can be evaluated using polar coordinates, with $x = r \cos \theta$ and $y = r \sin \theta$. The limits of integration become $0 \le r \le 5$ and $0 \le \theta \le \pi/2$. The surface area is then$$ \int_0^{\pi/2} \int_0^5 \sqrt{1 + 4r^2} \ r \ dr \ d\theta $$This integral can be evaluated using the substitution $u = 1 + 4r^2$, which gives $du = 8r \ dr$. The surface area is then$$ \int_0^{\pi/2} \int_1^{101} \frac{1}{8} \sqrt{u} \ du \ d\theta $$This integral can be evaluated using the power rule, which gives$$ \int_0^{\pi/2} \frac{1}{8} \left[ \frac{2}{3} u^{3/2} \right]_1^{101} \ d\theta $$The surface area is then$$ \int_0^{\pi/2} \frac{1}{12} (101^{3/2} - 1^{3/2}) \ d\theta $$This integral can be evaluated using the power rule, which gives$$ \frac{1}{12} \left[ \frac{2}{5} (101^{5/2} - 1^{5/2}) \right]_0^{\pi/2} $$The surface area is then$$ \frac{1}{12} \left( \frac{2}{5} (101^{5/2} - 1) \right) $$This simplifies to$$ \frac{1}{30} (101^{5/2} - 1) $$This is approximately equal to $10 \sin^{-1} \left(\frac{1}{5}\right)$.

The Surface Area of the Surface Cut From x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

The Surface Area of the Surface Cut From $x ^ { 2 } + y ^ { 2 } = 25$