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: -


13.PNG


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 x2 + y2 = a2 is equal to square units.


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