1. **State the problem:** Find the volume $V$ of the solid formed by rotating the region enclosed by the curves $y=5x^2+30$ and $y=40x-5$ about the y-axis. 2. **Find the points of intersection:** Set the two functions equal to find $x$ values where they intersect: $$5x^2 + 30 = 40x - 5$$ Rearranged: $$5x^2 - 40x + 35 = 0$$ Divide both sides by 5: $$\cancel{5}x^2 - \cancel{5}8x + \cancel{5}7 = 0 \Rightarrow x^2 - 8x + 7 = 0$$ 3. **Solve quadratic:** $$x = \frac{8 \pm \sqrt{64 - 28}}{2} = \frac{8 \pm \sqrt{36}}{2} = \frac{8 \pm 6}{2}$$ So, $$x=7 \quad \text{or} \quad x=1$$ 4. **Set up volume integral using the shell method:** When rotating around the y-axis, volume is $$V = 2\pi \int_{x=1}^{7} x \big(f(x) - g(x)\big) \, dx$$ where $f(x)$ is the upper curve and $g(x)$ the lower curve. 5. **Determine which function is upper:** At $x=1$, $y=5(1)^2+30=35$, $y=40(1)-5=35$ (equal) At $x=2$, $y=5(4)+30=50$, $y=80-5=75$ so $y=40x-5$ is upper. 6. **Write the integral:** $$V = 2\pi \int_1^7 x \big((40x - 5) - (5x^2 + 30)\big) dx = 2\pi \int_1^7 x (40x - 5 - 5x^2 - 30) dx$$ Simplify inside: $$40x - 5 - 5x^2 - 30 = -5x^2 + 40x - 35$$ 7. **Multiply by $x$:** $$x(-5x^2 + 40x - 35) = -5x^3 + 40x^2 - 35x$$ 8. **Integral becomes:** $$V = 2\pi \int_1^7 (-5x^3 + 40x^2 - 35x) dx$$ 9. **Integrate term-by-term:** $$\int (-5x^3) dx = -\frac{5x^4}{4}$$ $$\int 40x^2 dx = \frac{40x^3}{3}$$ $$\int (-35x) dx = -\frac{35x^2}{2}$$ 10. **Evaluate definite integral:** $$V = 2\pi \left[-\frac{5x^4}{4} + \frac{40x^3}{3} - \frac{35x^2}{2}\right]_1^7$$ Calculate at $x=7$: $$-\frac{5(7^4)}{4} + \frac{40(7^3)}{3} - \frac{35(7^2)}{2} = -\frac{5(2401)}{4} + \frac{40(343)}{3} - \frac{35(49)}{2} = -\frac{12005}{4} + \frac{13720}{3} - \frac{1715}{2}$$ Calculate at $x=1$: $$-\frac{5(1)}{4} + \frac{40(1)}{3} - \frac{35(1)}{2} = -\frac{5}{4} + \frac{40}{3} - \frac{35}{2}$$ 11. **Find difference:** $$\left(-\frac{12005}{4} + \frac{13720}{3} - \frac{1715}{2}\right) - \left(-\frac{5}{4} + \frac{40}{3} - \frac{35}{2}\right) = \left(-\frac{12005}{4} + \frac{13720}{3} - \frac{1715}{2}\right) + \frac{5}{4} - \frac{40}{3} + \frac{35}{2}$$ Simplify each term: $$-\frac{12005}{4} + \frac{5}{4} = -\frac{12000}{4} = -3000$$ $$\frac{13720}{3} - \frac{40}{3} = \frac{13680}{3} = 4560$$ $$-\frac{1715}{2} + \frac{35}{2} = -\frac{1680}{2} = -840$$ Sum: $$-3000 + 4560 - 840 = 720$$ 12. **Calculate final volume:** $$V = 2\pi \times 720 = 1440\pi$$ **Final answer:** $$\boxed{V = 1440\pi}$$