Question

Find the area of the smaller part of the circle x 2 + y 2 = a 2 cut off by the line

Answer 1

The given equation of circle is x 2 + y 2 = a 2 and the equation of the line is x= a 2 . Draw the graph of the circle and straight line and mark the region bounded by the circle and line with x-axis as ABDA.

Figure (1)

Mark a point C at the intersection of the line with the x-axis.

The area bounded by the curve and the x-axis.

To calculate the area of the region ABCA, we take a vertical strip in the region with infinitely small width, as shown in the figure above.

To find the area of the region ABCA, integrate the area of the strip.

Area of the region ABCA= ∫ a 2 a ydx (1)

The equation of the circle is x 2 + y 2 = a 2 . From this equation find the value of y in terms of x and substitute in equation (1).

x 2 + y 2 = a 2 y 2 = a 2 − x 2 y= a 2 − x 2

Substitute a 2 − x 2 for y in equation (1).

Area of the region ABCA= ∫ a 2 a a 2 − x 2 dx = [ x 2 ( ( a ) 2 − x 2 )+ ( a ) 2 2 sin −1 x a ] a 2 a =[ a 2 ( ( a ) 2 − a 2 )+ ( a ) 2 2 sin −1 a a −( ( a 2 2 ( a ) 2 − ( a 2 ) 2 )+ ( a ) 2 2 sin −1 a 2 a ) ] = a 2 2 ( π 2 )−( a 2 2 )( a 2 )+ a 2 2 π 4

Further simplify,

Area of region ACBA= a 2 4 ( π−1− π 2 ) = a 2 4 ( π 2 −1 )

Total area bounded by the circle and the straight line is twice of the area ACBA.

Total area=2×Area of region ACBA =2× a 2 4 ( π 2 −1 ) = a 2 2 ( π 2 −1 ) sq units

Thus, the required area is a 2 2 ( π 2 −1 ) sq units.