**Calculating the Volume of a Solid of Revolution**In this exercise, you are asked to find the volume of the solid generated by revolving a specific region about the x-axis.### Problem Statement:Find the volume of the solid generated by revolving the region bounded by the graphs of \(y = 2x^2 + 1\) and \(y = 2x + 14\) about the x-axis.---**Step-by-Step Solution**1. **Identify the functions involved:** - \(y = 2x^2 + 1\): This is a quadratic function representing a parabola opening upwards. - \(y = 2x + 14\): This is a linear function representing a straight line.2. **Find the intersection points of the functions:** - Set the equations equal to each other to find the points where the two graphs intersect: \[2x^2 + 1 = 2x + 14\] Simplifying this equation: \[2x^2 + 1 - 2x - 14 = 0\] \[2x^2 - 2x - 13 = 0\] Use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \(a = 2\), \(b = -2\), and \(c = -13\).3. **Calculate the volume:** - The volume \(V\) of the solid generated by revolving the region around the x-axis is found using the disk or washer method involving integration. - The volume formula using washers is: \[V = \pi \int_{a}^{b} \left[ (R(x))^2 - (r(x))^2 \right] dx\] where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius.4. **Determine the limits of integration and perform the integration:** - Use the intersection points as the limits of integration. - Integrate the squared functions of the boundaries to find the volume.### Exact Volume Expression:The volume of the solid generated by revolving the region bounded by the graphs of \(y = 2x^2 + 1\) and \(y = 2x + 14\) about the

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Find the volume of the solid generated by revolving the region bounded by the graphs of y = 2x² + 1 and y = 2x + 14 about the x-axis. C... The volume of the solid generated by revolving the region bounded by the graphs of y= 2x² + 1 and y = 2x + 14 about the x-axis is (Round to the nearest hundredth.) cubic units.

Find the volume of the solid generated by revolving the region bounded by the graphs of y = 2x² + 1 and y = 2x + 14 about the x-axis. C... The volume of the solid generated by revolving the region bounded by the graphs of y= 2x² + 1 and y = 2x + 14 about the x-axis is (Round to the nearest hundredth.) cubic units.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.5: Equations Of Lines

Problem 50E: The y-axis along with the graphs of y=-2x+7 and y=x+2 encloses a triangular region. Find the area of...

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