1. **State the problem:** We have the curve defined by $$y = 4\sqrt{x} - \frac{x}{2} + 1$$ for $$0 \leq x \leq 64$$. We need to find values of $$a, b, c$$ at $$x=16, 32, 48$$ respectively, estimate the shaded area using the trapezoidal rule, write the integral for the exact area, calculate the exact area, and find the percentage error of the estimate. 2. **Find values of a, b, c:** - At $$x=16$$: $$a = 4\sqrt{16} - \frac{16}{2} + 1 = 4 \times 4 - 8 + 1 = 16 - 8 + 1 = 9$$ - At $$x=32$$: $$b = 4\sqrt{32} - \frac{32}{2} + 1 = 4 \times \sqrt{32} - 16 + 1 = 4 \times 5.6569 - 15 = 22.6274 - 15 = 7.6274 \approx 7.63$$ - At $$x=48$$: $$c = 4\sqrt{48} - \frac{48}{2} + 1 = 4 \times 6.9282 - 24 + 1 = 27.7128 - 23 = 4.7128 \approx 4.71$$ 3. **Use trapezoidal rule to estimate area:** - Intervals: $$[0,16], [16,32], [32,48], [48,64]$$ each of width $$h=16$$. - Function values: $$y_0=1, y_1=9, y_2=7.63, y_3=4.71, y_4=1$$. - Trapezoidal rule formula: $$\text{Area} \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + y_3) + y_4]$$ - Calculate: $$= \frac{16}{2} [1 + 2(9 + 7.63 + 4.71) + 1] = 8 [1 + 2(21.34) + 1] = 8 [1 + 42.68 + 1] = 8 \times 44.68 = 357.44$$ 4. **Write down the integral for exact area:** $$\int_0^{64} \left(4\sqrt{x} - \frac{x}{2} + 1\right) dx$$ 5. **Calculate exact area using calculator:** - Compute integral: $$\int_0^{64} 4x^{1/2} - \frac{x}{2} + 1 \, dx = \left[ \frac{8}{3} x^{3/2} - \frac{x^2}{4} + x \right]_0^{64}$$ - Evaluate each term at $$x=64$$: $$\frac{8}{3} \times 64^{3/2} = \frac{8}{3} \times (64^{1/2})^3 = \frac{8}{3} \times 8^3 = \frac{8}{3} \times 512 = 1365.33$$ $$- \frac{64^2}{4} = - \frac{4096}{4} = -1024$$ $$+ 64 = 64$$ - Sum: $$1365.33 - 1024 + 64 = 405.33$$ - Rounded to one decimal place: $$405.3$$ 6. **Calculate percentage error:** $$\text{Error} = \frac{|\text{Estimate} - \text{Exact}|}{\text{Exact}} \times 100 = \frac{|357.44 - 405.33|}{405.33} \times 100 = \frac{47.89}{405.33} \times 100 \approx 11.82\%$$ **Final answers:** - $$a=9$$, $$b \approx 7.63$$, $$c \approx 4.71$$ - Trapezoidal estimate: $$357.44$$ - Exact area: $$405.3$$ - Percentage error: $$11.8\%$$