1. **State the problem:** We are given the marginal revenue function $$MR(x) = 4x^3 - x^2 + 3x^2 - 4$$ (in hundreds of dollars) for selling $$x$$ hand-built bicycles. We need to find the additional revenue earned when production increases from 3 to 7 bicycles. 2. **Understand the concept:** Marginal revenue $$MR(x)$$ represents the rate of change of total revenue $$R(x)$$ with respect to the number of bicycles $$x$$. To find the additional revenue when production increases from $$x=3$$ to $$x=7$$, we calculate the definite integral of $$MR(x)$$ from 3 to 7: $$\text{Additional Revenue} = \int_3^7 MR(x) \, dx$$ 3. **Simplify the marginal revenue function:** $$MR(x) = 4x^3 - x^2 + 3x^2 - 4 = 4x^3 + 2x^2 - 4$$ 4. **Integrate $$MR(x)$$:** $$\int MR(x) \, dx = \int (4x^3 + 2x^2 - 4) \, dx = \int 4x^3 \, dx + \int 2x^2 \, dx - \int 4 \, dx$$ Calculate each integral: $$\int 4x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4$$ $$\int 2x^2 \, dx = 2 \cdot \frac{x^3}{3} = \frac{2}{3}x^3$$ $$\int 4 \, dx = 4x$$ So, $$R(x) = x^4 + \frac{2}{3}x^3 - 4x + C$$ 5. **Calculate the additional revenue from 3 to 7:** $$\int_3^7 MR(x) \, dx = R(7) - R(3)$$ Calculate $$R(7)$$: $$7^4 + \frac{2}{3} \cdot 7^3 - 4 \cdot 7 = 2401 + \frac{2}{3} \cdot 343 - 28 = 2401 + \frac{686}{3} - 28$$ Calculate $$R(3)$$: $$3^4 + \frac{2}{3} \cdot 3^3 - 4 \cdot 3 = 81 + \frac{2}{3} \cdot 27 - 12 = 81 + 18 - 12 = 87$$ Now, $$R(7) = 2401 - 28 + \frac{686}{3} = 2373 + \frac{686}{3} = \frac{2373 \cdot 3}{3} + \frac{686}{3} = \frac{7119 + 686}{3} = \frac{7805}{3}$$ So, $$\text{Additional Revenue} = R(7) - R(3) = \frac{7805}{3} - 87 = \frac{7805}{3} - \frac{261}{3} = \frac{7544}{3} \approx 2514.67$$ 6. **Interpret the result:** The additional revenue earned when production increases from 3 to 7 bicycles is approximately 2514.67 hundreds of dollars, or 251,467 dollars.