1. **Problem statement:** We are given a function $$f(x) = -5 \cdot 10^{-5} \cdot x^4 + 0.06 \cdot x^2$$ and asked to compute two volumes: - $$V = \int_0^{60} f(x) \, dx$$ - $$V_{sand} = 2\pi \int_0^{30} x (30 - f(x)) \, dx$$ 2. **Formula and rules:** - The volume under the curve $$f(x)$$ from $$0$$ to $$60$$ is given by the definite integral $$V = \int_0^{60} f(x) \, dx$$. - The volume of the solid of revolution formed by rotating the curve $$y = 30 - f(x)$$ around the y-axis from $$0$$ to $$30$$ is given by $$V_{sand} = 2\pi \int_0^{30} x (30 - f(x)) \, dx$$. 3. **Calculate $$V$$:** $$V = \int_0^{60} \left(-5 \cdot 10^{-5} x^4 + 0.06 x^2\right) dx = \int_0^{60} -5 \cdot 10^{-5} x^4 \, dx + \int_0^{60} 0.06 x^2 \, dx$$ Calculate each integral: $$\int_0^{60} -5 \cdot 10^{-5} x^4 \, dx = -5 \cdot 10^{-5} \cdot \frac{x^5}{5} \Big|_0^{60} = -5 \cdot 10^{-5} \cdot \frac{60^5}{5}$$ $$\int_0^{60} 0.06 x^2 \, dx = 0.06 \cdot \frac{x^3}{3} \Big|_0^{60} = 0.06 \cdot \frac{60^3}{3}$$ Calculate values: $$-5 \cdot 10^{-5} \cdot \frac{60^5}{5} = -5 \cdot 10^{-5} \cdot \frac{777600000}{5} = -5 \cdot 10^{-5} \cdot 155520000 = -7776$$ $$0.06 \cdot \frac{60^3}{3} = 0.06 \cdot \frac{216000}{3} = 0.06 \cdot 72000 = 4320$$ Sum: $$V = -7776 + 4320 = -3456$$ 4. **Calculate $$V_{sand}$$:** $$V_{sand} = 2\pi \int_0^{30} x (30 - f(x)) \, dx = 2\pi \int_0^{30} (30x - x f(x)) \, dx$$ Substitute $$f(x)$$: $$x f(x) = x \left(-5 \cdot 10^{-5} x^4 + 0.06 x^2\right) = -5 \cdot 10^{-5} x^5 + 0.06 x^3$$ So: $$V_{sand} = 2\pi \int_0^{30} \left(30x - (-5 \cdot 10^{-5} x^5 + 0.06 x^3)\right) dx = 2\pi \int_0^{30} \left(30x + 5 \cdot 10^{-5} x^5 - 0.06 x^3\right) dx$$ Integrate term by term: $$\int_0^{30} 30x \, dx = 30 \cdot \frac{x^2}{2} \Big|_0^{30} = 15 \cdot 900 = 13500$$ $$\int_0^{30} 5 \cdot 10^{-5} x^5 \, dx = 5 \cdot 10^{-5} \cdot \frac{x^6}{6} \Big|_0^{30} = 5 \cdot 10^{-5} \cdot \frac{729000000}{6} = 5 \cdot 10^{-5} \cdot 121500000 = 6075$$ $$\int_0^{30} 0.06 x^3 \, dx = 0.06 \cdot \frac{x^4}{4} \Big|_0^{30} = 0.06 \cdot \frac{810000}{4} = 0.06 \cdot 202500 = 12150$$ Sum inside integral: $$13500 + 6075 - 12150 = 7425$$ Multiply by $$2\pi$$: $$V_{sand} = 2\pi \cdot 7425 = 14850 \pi \approx 46652.65092$$ 5. **Final answers:** $$V = -3456$$ $$V_{sand} \approx 46652.65092$$