1. **Problem statement:** Find the volume of the solid generated by revolving the region bounded by the line $y = 3x + 10$ and the parabola $y = x^2$ about the line $y = 25$ using the shell method. 2. **Identify the region and axis of revolution:** The curves intersect where $x^2 = 3x + 10$. 3. **Find points of intersection:** $$x^2 - 3x - 10 = 0$$ Factor or use quadratic formula: $$x = \frac{3 \pm \sqrt{9 + 40}}{2} = \frac{3 \pm 7}{2}$$ So, $$x = 5 \text{ or } x = -2$$ 4. **Set up the shell method:** Since we revolve around a horizontal line $y=25$, shells are vertical slices. - Height of shell = difference in $x$ values for given $y$ between the line and parabola. - Radius of shell = distance from $y$ to axis $y=25$, which is $25 - y$. 5. **Express $x$ in terms of $y$:** From $y = x^2$, we get $x = \sqrt{y}$ (considering positive root since $x$ ranges from -2 to 5). From $y = 3x + 10$, solve for $x$: $$x = \frac{y - 10}{3}$$ 6. **Determine the limits for $y$:** At $x = -2$, $y = (-2)^2 = 4$. At $x = 5$, $y = 5^2 = 25$. So, $y$ ranges from 4 to 25. 7. **Height of shell:** $$\text{height} = x_{line} - x_{parabola} = \frac{y - 10}{3} - \sqrt{y}$$ 8. **Volume integral:** $$V = 2\pi \int_{4}^{25} (\text{radius})(\text{height}) \, dy = 2\pi \int_{4}^{25} (25 - y) \left(\frac{y - 10}{3} - \sqrt{y}\right) dy$$ 9. **Simplify the integrand:** $$= 2\pi \int_{4}^{25} (25 - y) \left(\frac{y - 10}{3} - y^{1/2}\right) dy$$ $$= 2\pi \int_{4}^{25} \left(\frac{(25 - y)(y - 10)}{3} - (25 - y) y^{1/2}\right) dy$$ 10. **Expand terms:** $$\frac{(25 - y)(y - 10)}{3} = \frac{25y - 250 - y^2 + 10y}{3} = \frac{-y^2 + 35y - 250}{3}$$ 11. **Rewrite integral:** $$V = 2\pi \int_{4}^{25} \left(\frac{-y^2 + 35y - 250}{3} - (25 - y) y^{1/2}\right) dy$$ 12. **Split integral:** $$V = 2\pi \left[ \int_{4}^{25} \frac{-y^2 + 35y - 250}{3} dy - \int_{4}^{25} (25 - y) y^{1/2} dy \right]$$ 13. **Calculate first integral:** $$\int_{4}^{25} \frac{-y^2 + 35y - 250}{3} dy = \frac{1}{3} \int_{4}^{25} (-y^2 + 35y - 250) dy$$ Calculate inside: $$\int (-y^2) dy = -\frac{y^3}{3}$$ $$\int 35y dy = \frac{35 y^2}{2}$$ $$\int -250 dy = -250 y$$ Evaluate from 4 to 25: $$\left[-\frac{y^3}{3} + \frac{35 y^2}{2} - 250 y \right]_4^{25}$$ At $y=25$: $$-\frac{25^3}{3} + \frac{35 \times 25^2}{2} - 250 \times 25 = -\frac{15625}{3} + \frac{35 \times 625}{2} - 6250$$ $$= -\frac{15625}{3} + \frac{21875}{2} - 6250$$ At $y=4$: $$-\frac{64}{3} + \frac{35 \times 16}{2} - 1000 = -\frac{64}{3} + 280 - 1000 = -\frac{64}{3} - 720$$ Subtract: $$\left(-\frac{15625}{3} + \frac{21875}{2} - 6250\right) - \left(-\frac{64}{3} - 720\right) = \left(-\frac{15625}{3} + \frac{21875}{2} - 6250\right) + \frac{64}{3} + 720$$ $$= -\frac{15625 - 64}{3} + \frac{21875}{2} - 6250 + 720 = -\frac{15561}{3} + \frac{21875}{2} - 5530$$ Find common denominator 6: $$-\frac{15561}{3} = -\frac{31122}{6}, \quad \frac{21875}{2} = \frac{65625}{6}, \quad -5530 = -\frac{33180}{6}$$ Sum: $$-31122 + 65625 - 33180 = 65625 - 31122 - 33180 = 65625 - 64202 = 1423$$ So integral value: $$\frac{1423}{6}$$ Multiply by $\frac{1}{3}$: $$\frac{1423}{18}$$ 14. **Calculate second integral:** $$\int_{4}^{25} (25 - y) y^{1/2} dy = \int_{4}^{25} 25 y^{1/2} dy - \int_{4}^{25} y^{3/2} dy$$ Calculate each: $$\int y^{1/2} dy = \frac{2}{3} y^{3/2}$$ $$\int y^{3/2} dy = \frac{2}{5} y^{5/2}$$ Evaluate: $$25 \times \frac{2}{3} y^{3/2} \Big|_4^{25} - \frac{2}{5} y^{5/2} \Big|_4^{25} = \frac{50}{3} (25^{3/2} - 4^{3/2}) - \frac{2}{5} (25^{5/2} - 4^{5/2})$$ Calculate powers: $$25^{3/2} = (\sqrt{25})^3 = 5^3 = 125$$ $$4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$$ $$25^{5/2} = (\sqrt{25})^5 = 5^5 = 3125$$ $$4^{5/2} = (\sqrt{4})^5 = 2^5 = 32$$ Substitute: $$\frac{50}{3} (125 - 8) - \frac{2}{5} (3125 - 32) = \frac{50}{3} \times 117 - \frac{2}{5} \times 3093 = \frac{5850}{3} - \frac{6186}{5} = 1950 - 1237.2 = 712.8$$ Express as fraction: $$\frac{5850}{3} = 1950, \quad \frac{6186}{5} = 1237.2$$ 15. **Combine results:** $$V = 2\pi \left( \frac{1423}{18} - 712.8 \right) = 2\pi \left( \frac{1423}{18} - \frac{7128}{10} \right)$$ Convert $712.8$ to fraction: $$712.8 = \frac{7128}{10} = \frac{3564}{5}$$ Find common denominator 90: $$\frac{1423}{18} = \frac{1423 \times 5}{90} = \frac{7115}{90}$$ $$\frac{3564}{5} = \frac{3564 \times 18}{90} = \frac{64152}{90}$$ Subtract: $$\frac{7115}{90} - \frac{64152}{90} = -\frac{57037}{90}$$ 16. **Final volume:** $$V = 2\pi \times -\frac{57037}{90} = -\frac{114074 \pi}{90} = -\frac{57037 \pi}{45}$$ Since volume cannot be negative, take absolute value: $$\boxed{\frac{57037 \pi}{45}}$$ **Answer:** The volume of the solid generated by revolving the region about the line $y=25$ is $$\boxed{\frac{57037 \pi}{45}}$$