To find the area bounded by the curve y=ln(x) and the lines x=1 and x=4 above the x-axis, we can use the definite integral.
Step 1: Identify the limits of integration. In this case, the curve intersects the lines x=1 and x=4.
Step 2: Set up the integral. The area can be calculated by integrating the function ln(x) with respect to x between the limits of integration.
A = \int_{1}^{4} ln(x) \,dx
Step 3: Evaluate the integral. Integrate ln(x) using the integral of natural logarithm:
A = \Big[\int ln(x) \,dx\Big]_1^4 = \Big[xln(x) - x\Big]_1^4
Now, substitute the upper limit and the lower limit into the equation:
A = \Big[4ln(4) - 4\Big] - \Big[1ln(1) - 1\Big]
Step 4: Simplify the expression:
Since ln(1) = 0,
A=4ln(4)-4-0+1
Answer: The area bounded by the curve y=ln(x) and the lines x=1 and x=4 above the x-axis is 4ln(4)-3 .