1. **State the problem:** Evaluate the definite integral $$\int_4^8 \left(6t^{\frac{5}{2}} - 3t^{\frac{3}{2}}\right) dt$$ using the Fundamental Theorem of Calculus, part 2. 2. **Recall the theorem:** If $F(x)$ is an antiderivative of $f(x)$, then $$\int_a^b f(x) dx = F(b) - F(a).$$ 3. **Find the antiderivative:** For each term, use the power rule for integration: $$\int t^n dt = \frac{t^{n+1}}{n+1} + C$$ - For $6t^{\frac{5}{2}}$, the antiderivative is $$6 \cdot \frac{t^{\frac{5}{2}+1}}{\frac{5}{2}+1} = 6 \cdot \frac{t^{\frac{7}{2}}}{\frac{7}{2}} = 6 \cdot \frac{2}{7} t^{\frac{7}{2}} = \frac{12}{7} t^{\frac{7}{2}}.$$ - For $-3t^{\frac{3}{2}}$, the antiderivative is $$-3 \cdot \frac{t^{\frac{3}{2}+1}}{\frac{3}{2}+1} = -3 \cdot \frac{t^{\frac{5}{2}}}{\frac{5}{2}} = -3 \cdot \frac{2}{5} t^{\frac{5}{2}} = -\frac{6}{5} t^{\frac{5}{2}}.$$ So, $$F(t) = \frac{12}{7} t^{\frac{7}{2}} - \frac{6}{5} t^{\frac{5}{2}} + C.$$ 4. **Evaluate at the bounds:** $$F(8) = \frac{12}{7} \cdot 8^{\frac{7}{2}} - \frac{6}{5} \cdot 8^{\frac{5}{2}}, \quad F(4) = \frac{12}{7} \cdot 4^{\frac{7}{2}} - \frac{6}{5} \cdot 4^{\frac{5}{2}}.$$ Calculate powers: - $8^{\frac{7}{2}} = (8^{\frac{1}{2}})^7 = (\sqrt{8})^7 = (2\sqrt{2})^7 = 2^7 \cdot (\sqrt{2})^7 = 128 \cdot 2^{3.5} = 128 \cdot 11.3137 = 1447.68$ (approx) - $8^{\frac{5}{2}} = (\sqrt{8})^5 = (2\sqrt{2})^5 = 32 \cdot 5.6569 = 181.02$ (approx) - $4^{\frac{7}{2}} = (\sqrt{4})^7 = 2^7 = 128$ - $4^{\frac{5}{2}} = (\sqrt{4})^5 = 2^5 = 32$ 5. **Substitute and simplify:** $$F(8) = \frac{12}{7} \times 1447.68 - \frac{6}{5} \times 181.02 = 2483.88 - 217.22 = 2266.66$$ $$F(4) = \frac{12}{7} \times 128 - \frac{6}{5} \times 32 = 219.43 - 38.4 = 181.03$$ 6. **Calculate the definite integral:** $$\int_4^8 \left(6t^{\frac{5}{2}} - 3t^{\frac{3}{2}}\right) dt = F(8) - F(4) = 2266.66 - 181.03 = 2085.63.$$ **Final answer:** $$\boxed{2085.63}.$$