The area of the planar region bounded by the curve : and -axis
and the two straight lines equals

To find the area of the planar region bounded by the curve , the x-axis, and the lines and , we first need to set up the definite integral. The area can be calculated as:

Evaluating the integral:

Calculating at :

Calculating at :

Now, calculating the definite integral:

Thus, the area of the planar region is , which corresponds to option (a).

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The area we’re calculating represents the space above the x-axis and beneath the curve between the two vertical lines. It’s a fun exercise in integration! You get to see how math can help find “spaces” and understand shapes in a geometric context.

When approaching similar problems, always sketch the curves and lines you’re working with! Visualizing the area can help ensure you set the correct limits of integration and recognize where the curve lies relative to the x-axis. Common mistakes include forgetting to adjust for negative areas when curves dip below the axis or choosing incorrect bounds. Happy integrating!