Question

The area bounded by the curve y=ln(x) and the lines x=1 and x=4 above the x−axis is

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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 is4ln(4)-3 .

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.

Step 3: Evaluate the integral. Integrate ln(x) using the integral of natural logarithm:

Now, substitute the upper limit and the lower limit into the equation:

Step 4: Simplify the expression:

Since ln(1) = 0,

Answer: The area bounded by the curve y=ln(x) and the lines x=1 and x=4 above the x-axis is

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