Solve the problem by integration.
-Find the volume generated by revolving the first-quadrant area bounded by $y = \frac { 6 } { x ^ { 4 } + 4 x ^ { 2 } + 3 }$ and $x = 5$ about the $y$-axis.
A) $3 \pi \ln \frac { 39 } { 14 }$
B) $3 \pi \ln 2184$
C) $\frac { 3 } { 2 } \pi \ln 2184$
D) $6 \pi \ln \frac { 39 } { 14 }$

Question

Quizplus · Accepted Answer

The volume of the solid generated by revolving the area bounded by $y = \frac { 6 } { x ^ { 4 } + 4 x ^ { 2 } + 3 }$ and $x = 5$ about the $y$-axis can be found using the method of cylindrical shells. The formula for the volume of a solid of revolution generated by revolving a function $y = f(x)$ about the $y$-axis from $x = a$ to $x = b$ is given by $V = 2\pi \int_{a}^{b} x f(x) dx$.Given $y = \frac { 6 } { x ^ { 4 } + 4 x ^ { 2 } + 3 }$, we revolve this around the $y$-axis from $x = 0$ to $x = 5$. Thus, the volume $V$ is:$$V = 2\pi \int_{0}^{5} x \left( \frac { 6 } { x ^ { 4 } + 4 x ^ { 2 } + 3 } \right) dx$$Let's simplify the integral:$$V = 2\pi \int_{0}^{5} \frac { 6x } { x ^ { 4 } + 4 x ^ { 2 } + 3 } dx$$Notice that the denominator can be rewritten as $(x^2+3)(x^2+1)$, making it easier to integrate using partial fractions or recognizing it as a standard integral form. However, the key insight is that the integral simplifies to a form involving a natural logarithm, specifically because the derivative of the denominator matches the numerator (after factoring out constants).After performing the integration, the result is $3 \pi \ln \frac { 39 } { 14 }$, which matches choice A. This is because the integral evaluates to a logarithmic function of the limits of integration, and after simplification, it yields the given answer.

Solve the Problem by Integration y=6x4+4x2+3y = \frac { 6 } { x ^ { 4 } + 4 x ^ { 2 } + 3 }y=x4+4x2+36​

Solve the Problem by Integration $y = \frac { 6 } { x ^ { 4 } + 4 x ^ { 2 } + 3 }$