To evaluate the areas of bounded regions, one has to know the rough sketch of the curve.
The principles/ideas that help in visualizing the curve or rough sketching the curve are:
1. Origin and tangents at the origin
Check whether the curve passes through the origin and finding out whether (0,0) satisfies the curve equation.
If the curve passes through (0,0) find the equation of tangent at (0,0)
2. Points of intersection of the curve with the coordinate axes
You can find the point at which the curve crosses the x-axis by putting y = 0 in the curve equation.
Similarly putting x = 0 gives the point at which the curve passes y-axis.
If all powers of y in the curve equation are even, then the curve is symmetric about x-axis.
The familiar curve y² = 4ax is symmetric about x-axis.
If all powers x in the equation of curve are even, it is symmetric about y-axis.
If by putting x = -x and y = -y, the equation of the curve remains the same, then it symmetric in opposite directions.
If the equation remains unaltered by interchanging x and y in the equation, then it is symmetric about the line y =x.
4. Regions where the curve does not exist
In the regions where the curve does not exist, x or y values will be imaginary.
In the case of y² = 4ax, when x is negative, y values will be imaginary. Hence curve does not exist on the left side of y-axis.
5. Find points where dy/dx = 0
At these points, the slope of the tangent to the curve is zero and the tangent will be parallel to the x axis.
6. Find maxima and minima points
Find points where dy/dx = 0 and check the sign of d²y/dx² at these points and find out maximum (d²y/dx² is negative) and minimum (d²y/dx² is positive) points of the curve.
Also find the interval in which the dy/dx is g.t. 0. In this interval, the function is monotonically increasing.
In the interval in which dy/dx is l.t. 0, the function is monotonically decreasing.