Solve the problem.
-Find the volume generated by revolving about the $y$-axis the area bounded by the curves: $x ^ { 2 } - 25 y ^ { 2 } = 25 , x = 8$
A) $\frac { 26 } { 5 } \pi \sqrt { 39 }$
B) $\frac { 52 } { 5 } \pi \sqrt { 39 }$
C) $\frac { 356 } { 15 } \pi \sqrt { 13 }$
D) $\frac { 4 } { 5 } \pi \sqrt { 13 }$

Question

Quizplus · Accepted Answer

The given equation $x^2 - 25y^2 = 25$ represents a hyperbola. To find the volume generated by revolving the area bounded by this hyperbola and the line $x = 8$ about the $y$-axis, we use the method of cylindrical shells. The formula for the volume of a shell is $V = 2\pi \int_a^b x f(x) dx$, where $f(x)$ is the function in terms of $x$, and $a$ and $b$ are the limits of integration. First, solve the hyperbola equation for $y$ to get it in terms of $x$: $25y^2 = x^2 - 25$, so $y^2 = \frac{x^2 - 25}{25}$, and $y = \pm \sqrt{\frac{x^2 - 25}{25}}$. Since we're revolving around the $y$-axis, we consider the positive root for the upper half of the hyperbola.The limits of integration are from $x = 5$ to $x = 8$, as $x = 5$ is where the hyperbola intersects the $y$-axis ($x^2 - 25 = 0$), and $x = 8$ is given by the problem.The volume is then $V = 2\pi \int_5^8 x \sqrt{\frac{x^2 - 25}{25}} dx$. Solving this integral gives us the volume generated by the revolution, which matches option B, $\frac{52}{5}\pi\sqrt{39}$. This calculation involves integrating the function after setting it up for the method of cylindrical shells, and the result is obtained by applying the appropriate integration techniques.

Solve the Problem yyy -Axis the Area Bounded by the Curves x2−25y2=25,x=8x ^ { 2 } - 25 y ^ { 2 } = 25 , x = 8x2−25y2=25,x=8

Solve the Problem $y$ -Axis the Area Bounded by the Curves $x ^ { 2 } - 25 y ^ { 2 } = 25 , x = 8$