Using definite integrals, find the area of circle x^{2} + y^{2} = a^{2}

Given equations are :

x^{2} + y^{2} = a^{2} ...... (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 )

Substitute

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 x^{2} + y^{2} = a^{2} is equal to square units.

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