Using definite integrals, find the area of circle x2 + y2 = a2

Given equations are :

x2 + y2 = a2 ...... (1)

Equation (1) represents a circle with centre (0,0) and radius a, so it meets the axes at (±a,0), (0,±a). A rough sketch of the curve is given below: -


We have to find the area of shaded region.

Required area

= (shaded region ABCDA)

= 4(shaded region OBCO) (as the circle is symmetrical about the x - axis as well as the y - axis)

(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)

(As x is between (0,a) and the value of y varies)

(as )


So the above equation becomes,

We know,

So the above equation becomes,

Apply reduction formula:

On integrating we get,

Undo the substituting, we get

On applying the limits we get,

Hence the area of circle x2 + y2 = a2 is equal to square units.