1. **State the problem:** We have the curve $y = (x - 2)(x - 4)$ and a vertical line $x = k$ with $k > 4$. The total shaded area between the curve and the x-axis from $x=2$ to $x=4$ and from $x=4$ to $x=k$ is given as $\frac{58}{3}$. We need to find the value of $k$. 2. **Analyze the curve:** The curve is $y = (x - 2)(x - 4) = x^2 - 6x + 8$. It crosses the x-axis at $x=2$ and $x=4$. 3. **Calculate the area from $x=2$ to $x=4$:** Since the curve is below the x-axis between 2 and 4 (because the parabola opens upwards and the roots are at 2 and 4), the area is: $$A_1 = \int_2^4 |(x-2)(x-4)| \, dx = -\int_2^4 (x^2 - 6x + 8) \, dx$$ Calculate the integral: $$\int_2^4 (x^2 - 6x + 8) \, dx = \left[ \frac{x^3}{3} - 3x^2 + 8x \right]_2^4$$ Evaluate at 4: $$\frac{4^3}{3} - 3(4^2) + 8(4) = \frac{64}{3} - 48 + 32 = \frac{64}{3} - 16 = \frac{64 - 48}{3} = \frac{16}{3}$$ Evaluate at 2: $$\frac{2^3}{3} - 3(2^2) + 8(2) = \frac{8}{3} - 12 + 16 = \frac{8}{3} + 4 = \frac{8 + 12}{3} = \frac{20}{3}$$ So the integral is: $$\frac{16}{3} - \frac{20}{3} = -\frac{4}{3}$$ Therefore, $$A_1 = -(-\frac{4}{3}) = \frac{4}{3}$$ 4. **Calculate the area from $x=4$ to $x=k$:** For $x > 4$, the curve is positive (above x-axis), so the area is: $$A_2 = \int_4^k (x^2 - 6x + 8) \, dx = \left[ \frac{x^3}{3} - 3x^2 + 8x \right]_4^k$$ Evaluate at $k$: $$\frac{k^3}{3} - 3k^2 + 8k$$ Evaluate at 4 (already calculated): $$\frac{64}{3} - 48 + 32 = \frac{16}{3}$$ So, $$A_2 = \left( \frac{k^3}{3} - 3k^2 + 8k \right) - \frac{16}{3}$$ 5. **Total area given:** $$A_1 + A_2 = \frac{4}{3} + \left( \frac{k^3}{3} - 3k^2 + 8k - \frac{16}{3} \right) = \frac{58}{3}$$ Simplify: $$\frac{4}{3} - \frac{16}{3} = -\frac{12}{3} = -4$$ So, $$\frac{k^3}{3} - 3k^2 + 8k - 4 = \frac{58}{3}$$ Multiply both sides by 3: $$k^3 - 9k^2 + 24k - 12 = 58$$ Bring all terms to one side: $$k^3 - 9k^2 + 24k - 70 = 0$$ 6. **Solve the cubic equation:** Try possible integer roots using Rational Root Theorem: factors of 70 are ±1, ±2, ±5, ±7, ±10, ±14, ±35, ±70. Test $k=5$: $$5^3 - 9(5^2) + 24(5) - 70 = 125 - 225 + 120 - 70 = (125 - 225) + (120 - 70) = -100 + 50 = -50 \neq 0$$ Test $k=7$: $$7^3 - 9(7^2) + 24(7) - 70 = 343 - 441 + 168 - 70 = (343 - 441) + (168 - 70) = -98 + 98 = 0$$ So, $k=7$ is a root. 7. **Answer:** $$\boxed{7}$$ The value of $k$ is 7.