1. **State the problem:** Find the area under the curve $y=\frac{11}{x}$ above the x-axis over the interval $[1,2]$. 2. **Set up the integral:** The area $A$ is given by the definite integral of the function from 1 to 2: $$A=\int_1^2 \frac{11}{x} \, dx$$ 3. **Find the antiderivative:** Recall that the antiderivative of $\frac{1}{x}$ is $\ln|x|$. Therefore, $$\int \frac{11}{x} \, dx = 11 \ln|x| + C$$ For the definite integral, we do not include the constant $C$. 4. **Evaluate the definite integral:** $$A = 11 \ln|x| \Big|_1^2 = 11 (\ln 2 - \ln 1)$$ Since $\ln 1 = 0$, this simplifies to $$A = 11 \ln 2$$ 5. **Calculate the numerical value:** Using $\ln 2 \approx 0.693$, $$A \approx 11 \times 0.693 = 7.623$$ **Final answer:** The area under the curve is approximately $7.623$ units$^2$.