1. **State the problem:** Find the area under the curve $y = 2\sqrt{x}$ from $x=2$ to $x=7$. 2. **Formula used:** The area under a curve $y=f(x)$ from $x=a$ to $x=b$ is given by the definite integral: $$\text{Area} = \int_a^b f(x)\,dx$$ 3. **Apply the formula:** Here, $f(x) = 2\sqrt{x} = 2x^{1/2}$, so $$\text{Area} = \int_2^7 2x^{1/2} \, dx$$ 4. **Integrate:** Use the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ So, $$\int 2x^{1/2} \, dx = 2 \int x^{1/2} \, dx = 2 \cdot \frac{x^{3/2}}{3/2} = 2 \cdot \frac{2}{3} x^{3/2} = \frac{4}{3} x^{3/2}$$ 5. **Evaluate the definite integral:** $$\text{Area} = \left[ \frac{4}{3} x^{3/2} \right]_2^7 = \frac{4}{3} \left(7^{3/2} - 2^{3/2} \right)$$ 6. **Calculate powers:** $$7^{3/2} = (\sqrt{7})^3 = (2.6458)^3 \approx 18.520$$ $$2^{3/2} = (\sqrt{2})^3 = (1.4142)^3 \approx 2.828$$ 7. **Substitute and simplify:** $$\text{Area} = \frac{4}{3} (18.520 - 2.828) = \frac{4}{3} \times 15.692 = \frac{4 \times 15.692}{3}$$ 8. **Final calculation:** $$\text{Area} = \frac{62.768}{3} \approx 20.923$$ **Answer:** The area under the curve $y=2\sqrt{x}$ from $x=2$ to $x=7$ is approximately **20.923** (rounded to three decimal places).