Discussion
The bounded region enclosed by the curve \(y = x^{2} + 2\) and the line \(y = 4x - 1\) is revolved about the \(x\)-axis to generate a solid. The volume of the generated solid equals
Answer: A
The volume is \(\frac{88}{5}\pi\).
The curves meet where \(x^{2} + 2 = 4x - 1\), that is \(x^{2} - 4x + 3 = 0\), so \(x = 1\) and \(x = 3\). Between them the line lies above the parabola (at \(x = 2\): 7 against 6), and both are positive, so the washer method gives
\[ \begin{aligned} V = \pi\int_{1}^{3} \Big[ &(4x - 1)^{2} \\ &- (x^{2} + 2)^{2} \Big]\, dx \end{aligned} \]
\[ \begin{aligned} &= \pi\int_{1}^{3} (-x^{4} + 12x^{2} - 8x - 3)\,dx \\ &= \pi\left(-\frac{242}{5} + 104 - 32 - 6\right) \\ &= \pi\left(-\frac{242}{5} + 66\right) = \frac{88}{5}\pi \end{aligned} \]
The \(x^{4}\) term is the one to watch: it contributes \(-\frac{3^{5} - 1}{5} = -\frac{242}{5}\). The other options, \(\frac{84}{5}\pi\), \(\frac{82}{5}\pi\) and \(\frac{92}{5}\pi\), do not match the integral; note also that revolving the region needs the difference of the squares, not the square of the difference.
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