Integral

Step 1: Express revenue as a function of
Revenue is given by:
Step 2: Differentiate to find critical points
The first derivative of is: For maximum or minimum, set :
Step 3: Verify using the second derivative
The second derivative of is: Since , is maximum when .
Step 4: Final result
The number of units that should be sold to maximise revenue is .

Step 1: Rewrite the function
The function can be expressed as:
Step 2: Graph the function
The graph of is a parabola, concave downwards for and concave upwards for . (Refer to the attached graph.)

Step 3: Area computation using integration
The area of the shaded region between and is given by:
Step 4: Evaluate the integral
Thus, the total area is: Step 5: Final result
The area bounded by the curve , the X-axis, and the ordinates and is: