1. **State the problem:** Estimate the integral $$\int_5^7 2^{5 - \sqrt{x}} \, dx$$ using the trapezium rule with the given values of $y$ at $x = 5, 5.5, 6, 6.5, 7$. 2. **Recall the trapezium rule formula:** For equally spaced points $x_0, x_1, \ldots, x_n$ with spacing $h$, the trapezium rule approximation is: $$\int_{x_0}^{x_n} f(x) \, dx \approx \frac{h}{2} \left(y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n\right)$$ 3. **Identify values:** - $x$ values: 5, 5.5, 6, 6.5, 7 - $y$ values: 6.792, 6.298, 5.858, 5.466, 5.113 - Number of intervals $n=4$ - Step size $h = 5.5 - 5 = 0.5$ 4. **Apply trapezium rule:** $$\int_5^7 2^{5 - \sqrt{x}} \, dx \approx \frac{0.5}{2} \left(6.792 + 2(6.298 + 5.858 + 5.466) + 5.113\right)$$ Calculate the sum inside parentheses: $$6.792 + 2(6.298 + 5.858 + 5.466) + 5.113 = 6.792 + 2(17.622) + 5.113 = 6.792 + 35.244 + 5.113 = 47.149$$ So, $$\int_5^7 2^{5 - \sqrt{x}} \, dx \approx \frac{0.5}{2} \times 47.149 = 0.25 \times 47.149 = 11.78725$$ Rounded to 2 decimal places: $$\boxed{11.79}$$ --- 5. **Part (b)(i): Estimate $$\int_5^7 2^{6 - \sqrt{x}} \, dx$$** Note that: $$2^{6 - \sqrt{x}} = 2^{1} \times 2^{5 - \sqrt{x}} = 2 \times 2^{5 - \sqrt{x}}$$ Therefore, $$\int_5^7 2^{6 - \sqrt{x}} \, dx = \int_5^7 2 \times 2^{5 - \sqrt{x}} \, dx = 2 \times \int_5^7 2^{5 - \sqrt{x}} \, dx$$ Using the answer from part (a): $$= 2 \times 11.78725 = 23.5745$$ Rounded to 2 decimal places: $$\boxed{23.57}$$ --- 6. **Part (b)(ii): Estimate $$\int_5^7 (3 + 2^{5 - \sqrt{x}}) \, dx$$** Split the integral: $$\int_5^7 3 \, dx + \int_5^7 2^{5 - \sqrt{x}} \, dx$$ Calculate the first integral: $$3 \times (7 - 5) = 3 \times 2 = 6$$ Add the result from part (a): $$6 + 11.78725 = 17.78725$$ Rounded to 2 decimal places: $$\boxed{17.79}$$