1. **Problem Statement:** Find the value of $\sqrt{2.22}$ given that $\sqrt{1.8} = 1.341$. 2. **Given Information and Formula:** We know $\sqrt{1.8} = 1.341$. We want to estimate $\sqrt{2.22}$. 3. **Approach:** Since $2.22$ is close to $1.8$, we can use a linear approximation or the binomial expansion for square roots. 4. **Using the binomial approximation for square roots:** For $\sqrt{a + h} \approx \sqrt{a} + \frac{h}{2\sqrt{a}}$ when $h$ is small. 5. **Apply the formula:** Let $a = 1.8$, $h = 2.22 - 1.8 = 0.42$. $$\sqrt{2.22} \approx \sqrt{1.8} + \frac{0.42}{2 \times 1.341}$$ 6. **Calculate:** $$\sqrt{2.22} \approx 1.341 + \frac{0.42}{2.682} = 1.341 + 0.1567 = 1.4977$$ 7. **Final answer:** $$\boxed{\sqrt{2.22} \approx 1.498}$$ --- **Note on the integral expressions and calculations you provided:** - The integral expressions and their evaluations seem unrelated to the square root problem. - The integral formula $\int x^n dx = \frac{x^{n+1}}{n+1}$ is correct for $n \neq -1$. - The integral evaluations you wrote contain inconsistent terms and limits, and the steps are unclear. - Therefore, the integral part appears incorrect or incomplete as presented. Hence, the square root approximation is correct and the integral problem as stated is not correctly solved or explained here.