RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 27 of Exercise 21.1

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27

Find the area of the minor segment of the circle x2 + y2 = a2 cut off by the line .

Given equations are :

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


..... (2)


Equation (1) represents a circle with centre (0,0) and radius a, so it meets the axes at (±a,0), (0,±a).


Equation (2) represents a line parallel to y axis.


A rough sketch of the circle is given below: -


13.PNG


We have to find the area of shaded region.


Required area


= (shaded region BCDB)


= 2(shaded region ABCA) (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 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 the minor segment of the circle x2 + y2 = a2 cut off by the line is equal to square units.


27

Chapter Exercises

More Exercise Questions

7

Sketch the graph of in [0,4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4