Solve the problem.
-Find the volume generated by revolving about the $y$-axis the area bounded by the curves $2 x ^ { 2 } - y ^ { 2 } = 50 , x = 8$
A) $52 \pi \sqrt { 39 }$
B) $26 \pi \sqrt { 78 }$
C) $52 \pi \sqrt { 78 }$
D) $52 \pi \sqrt { 26 }$

Question

Quizplus · Accepted Answer

The volume of the solid of revolution can be found using the method of cylindrical shells. The given equation can be rewritten as $y^2 = 2x^2 - 50$. The limits of integration for $x$ are from the point where $y = 0$ to $x = 8$. Solving for $x$ when $y = 0$, we get $x = \sqrt{25} = 5$. The volume is calculated as $V = 2\pi \int_{5}^{8} x \sqrt{2x^2 - 50} \, dx$, which evaluates to $52 \pi \sqrt{78}$.

Solve the Problem yyy -Axis the Area Bounded by the Curves 2x2−y2=50,x=82 x ^ { 2 } - y ^ { 2 } = 50 , x = 82x2−y2=50,x=8

Solve the Problem $y$ -Axis the Area Bounded by the Curves $2 x ^ { 2 } - y ^ { 2 } = 50 , x = 8$