Step 1: Identify the boundaries of the region.
The region is bounded by the curve (which is ), the x-axis ( ), and the vertical line .
We need to find the point where the curve intersects the x-axis to determine the lower limit of integration. Set :
So, the region is bounded by , , , and . Step 2: Set up the definite integral for the area.
The area under the curve from to above the x-axis is given by .
Here, , , and . Since for , the area is:
Step 3: Evaluate the indefinite integral using integration by parts.
Let and . Then and .
Using the formula :
Step 4: Evaluate the definite integral using the limits of integration.
Substitute and :
Step 5: Compare the result with the given options.
The calculated area is 1, which matches option (C).