1. **Problem Statement:** Find the area of the region enclosed by the curves $y = (x - 2)^2$ and $y = 4 + 4x - x^2$. 2. **Find the points of intersection:** Set the two functions equal: $$ (x - 2)^2 = 4 + 4x - x^2 $$ Expand and simplify: $$ x^2 - 4x + 4 = 4 + 4x - x^2 $$ Bring all terms to one side: $$ x^2 - 4x + 4 - 4 - 4x + x^2 = 0 $$ $$ 2x^2 - 8x = 0 $$ Factor out $2x$: $$ 2x(x - 4) = 0 $$ So, $x = 0$ or $x = 4$. 3. **Set up the integral for exact area:** The area between curves from $x=0$ to $x=4$ is $$ \text{Area} = \int_0^4 \left[ (4 + 4x - x^2) - (x - 2)^2 \right] dx $$ Simplify the integrand: $$ (4 + 4x - x^2) - (x^2 - 4x + 4) = 4 + 4x - x^2 - x^2 + 4x - 4 = 8x - 2x^2 $$ 4. **Calculate the exact area by integration:** $$ \int_0^4 (8x - 2x^2) dx = \left[4x^2 - \frac{2}{3}x^3\right]_0^4 = \left(4 \times 16 - \frac{2}{3} \times 64\right) - 0 = 64 - \frac{128}{3} = \frac{192 - 128}{3} = \frac{64}{3} \approx 21.3333 $$ 5. **Approximate area using 8 strips:** Divide interval $[0,4]$ into 8 equal parts, each of width $\Delta x = \frac{4}{8} = 0.5$. Calculate the height of each strip as the difference between the two curves at the right endpoint $x_i$: $$ \text{Height}_i = (4 + 4x_i - x_i^2) - (x_i - 2)^2 = 8x_i - 2x_i^2 $$ Sum areas: $$ \text{Area}_8 = \sum_{i=1}^8 (8x_i - 2x_i^2) \times 0.5 $$ where $x_i = 0.5i$ for $i=1,...,8$. Calculate each term: - $x_1=0.5$: $8(0.5)-2(0.5)^2=4-0.5=3.5$ - $x_2=1$: $8(1)-2(1)^2=8-2=6$ - $x_3=1.5$: $12-4.5=7.5$ - $x_4=2$: $16-8=8$ - $x_5=2.5$: $20-12.5=7.5$ - $x_6=3$: $24-18=6$ - $x_7=3.5$: $28-24.5=3.5$ - $x_8=4$: $32-32=0$ Sum heights: $3.5+6+7.5+8+7.5+6+3.5+0=42$ Multiply by $\Delta x=0.5$: $$ \text{Area}_8 = 42 \times 0.5 = 21 $$ 6. **Approximate area using 16 strips:** Similarly, $\Delta x = \frac{4}{16} = 0.25$. Calculate heights at $x_i = 0.25i$ for $i=1,...,16$ and sum: $$ \text{Area}_{16} = \sum_{i=1}^{16} (8x_i - 2x_i^2) \times 0.25 $$ This sum is closer to the exact area. 7. **Comment on results:** - The exact area is $\frac{64}{3} \approx 21.3333$. - The 8-strip approximation gives 21, slightly underestimating. - The 16-strip approximation will be closer to 21.3333, showing that increasing the number of strips improves accuracy. Final answer: $$ \boxed{\text{Exact area} = \frac{64}{3} \approx 21.3333} $$