1. **State the problem:** We have the curve $$y = 3 + 2x - x^2$$ and a point $$A$$ on the curve at $$x=1.5$$. The tangent at $$A$$ meets the x-axis at $$B$$, and the curve meets the x-axis at $$C$$. We want to find the area of the shaded region bounded by the curve from $$A$$ to $$C$$ and the tangent line from $$A$$ to $$B$$. 2. **Find coordinates of points:** - At $$A$$, $$x=1.5$$, so $$y_A = 3 + 2(1.5) - (1.5)^2 = 3 + 3 - 2.25 = 3.75$$. - To find $$C$$, solve $$3 + 2x - x^2 = 0$$: $$x^2 - 2x - 3 = 0$$ Factor: $$(x-3)(x+1) = 0$$ So $$x=3$$ or $$x=-1$$. Since $$A$$ is at $$1.5$$, $$C$$ is at $$x=3$$. 3. **Find the tangent line at $$A$$:** - Derivative of $$y$$ is $$y' = 2 - 2x$$. - At $$x=1.5$$, slope $$m = 2 - 2(1.5) = 2 - 3 = -1$$. - Equation of tangent line at $$A(1.5, 3.75)$$: $$y - 3.75 = -1(x - 1.5)$$ $$y = -x + 1.5 + 3.75 = -x + 5.25$$. 4. **Find point $$B$$ where tangent meets x-axis:** - Set $$y=0$$ in tangent line: $$0 = -x + 5.25 \\ x = 5.25$$ - So $$B = (5.25, 0)$$. 5. **Set up the integral for the shaded area:** - The shaded area is between $$x=1.5$$ and $$x=3$$ bounded by the curve and tangent. - The curve is below the tangent line in this interval, so area = $$\int_{1.5}^{3} (\text{tangent} - \text{curve}) \, dx$$. 6. **Write the integrand:** - Tangent line: $$y_t = -x + 5.25$$ - Curve: $$y_c = 3 + 2x - x^2$$ - Difference: $$y_t - y_c = (-x + 5.25) - (3 + 2x - x^2) = -x + 5.25 - 3 - 2x + x^2 = x^2 - 3x + 2.25$$. 7. **Calculate the integral:** $$\int_{1.5}^{3} (x^2 - 3x + 2.25) \, dx = \left[ \frac{x^3}{3} - \frac{3x^2}{2} + 2.25x \right]_{1.5}^{3}$$ Evaluate at $$x=3$$: $$\frac{27}{3} - \frac{3 \times 9}{2} + 2.25 \times 3 = 9 - 13.5 + 6.75 = 2.25$$ Evaluate at $$x=1.5$$: $$\frac{(1.5)^3}{3} - \frac{3 (1.5)^2}{2} + 2.25 \times 1.5 = \frac{3.375}{3} - \frac{3 \times 2.25}{2} + 3.375 = 1.125 - 3.375 + 3.375 = 1.125$$ Subtract: $$2.25 - 1.125 = 1.125$$ 8. **Final answer:** The area of the shaded region is $$1.125$$ square units.