Solved Examples

Example 1:

   Find the area common to the curves x² + y² = 4x and y² = x.

Solution:

x² + y² = 4x ……… (i)

(x – 2)² + y² = 4

   common-area

   This is a circle with centre at (2, 0) and radius 2.

   y = √(4x-x² )

   y² = x ……… (ii)

Parabola with vertex as origin and symmetrical about x-axis. We will find the area above the x-axis and double the area.

   The two curves intersect at 4x – x² = x

x² – 3x = 0

x(x – 3) = 0

x = 0, 3

⇒ y = 0, ± √3

Therefore pts. are (0, 0) and (3, ± √3)

The required area is

Example 2:

Find the area enclosed between the curve y2 = 4ax and parabola x² = 4by

Solution:

For points of intersection solving the two equations:

   y² = 4ax

x² = 4by

   ⇒ (x²/4b)² = 4ax

   x = 0 or x = ³√(64ab²)

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