Question Let \( R \) represent the region bounded above by the graph of \( f(x)=2 x+7 \), below by the graph of \( g(x)=\frac{6}{x} \), and on eac side by \( x=1 \) and \( x=4 \). What is the volume of the solid of revolution obtained by rotating \( R \) about the \( x \)-axis? Enter an exact value in terms of \( \pi \).

To find the volume of the solid of revolution obtained by rotating the region \( R \) around the \( x \)-axis, we can use the method of washers. The volume \( V \) is given by the integral \[ V = \pi \int_{a}^{b} \left( f(x)^2 - g(x)^2 \right) \, dx \] In our case, \( f(x) = 2x + 7 \) and \( g(x) = \frac{6}{x} \), with the limits of integration being from \( x = 1 \) to \( x = 4 \). Let's calculate the volume step by step: 1. Compute \( f(x)^2 \) and \( g(x)^2 \): \[ f(x)^2 = (2x + 7)^2 = 4x^2 + 28x + 49 \] \[ g(x)^2 = \left(\frac{6}{x}\right)^2 = \frac{36}{x^2} \] 2. Set up the integral: \[ V = \pi \int_{1}^{4} \left( (4x^2 + 28x + 49) - \frac{36}{x^2} \right) \, dx \] 3. Combine the terms: \[ V = \pi \int_{1}^{4} \left( 4x^2 + 28x + 49 - \frac{36}{x^2} \right) \, dx \] 4. Evaluate the integral. We compute it term by term: \[ \int 4x^2 \, dx = \frac{4}{3}x^3, \quad \int 28x \, dx = 14x^2, \quad \int 49 \, dx = 49x, \quad \int \frac{36}{x^2} \, dx = -\frac{36}{x} \] 5. Now substituting the limits \( x = 1 \) and \( x = 4 \): \[ V = \pi \left[ \left( \frac{4}{3}(4^3) + 14(4^2) + 49(4) + \frac{36}{1} \right) - \left( \frac{4}{3}(1^3) + 14(1^2) + 49(1) + \frac{36}{1} \right) \right] \] 6. Calculate: \[ \frac{4}{3}(64) + 14(16) + 49(4) - \left( \frac{4}{3}(1) + 14(1) + 49(1) \right) = \frac{256}{3} + 224 + 196 - \left( \frac{4}{3} + 14 + 49 \right) \] Combine everything: \[ \frac{256}{3} + 224 + 196 = \frac{256}{3} + \frac{672}{3} + \frac{588}{3} = \frac{1516}{3} \] On the left for x=1: \[ \frac{4}{3} + 14 + 49 = \frac{4}{3} + 63 = \frac{193}{3} \] Now, \[ V = \pi \left( \frac{1516}{3} - \frac{193}{3} \right) = \pi \left( \frac{1323}{3} \right) = 441\pi \] Thus, the volume of the solid of revolution is \[ \boxed{441\pi} \]