RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 11 of Exercise 21.1

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11

Sketch the region {(x,y):9x2 + 4y2 = 36} and the find the area of the region enclosed by it, using integration

Given equation:

9x2 + 4y2 = 36 ...... (1)


equation (1) represents an eclipse that is symmetrical about the x - axis and also about the y - axis, with center at origin and passes through (±2, 0) and (0, ±3).


A rough sketch is given as below: -


9.PNG


We have to find the area of the shaded region.


Required area


= shaded region ABCDA


= 4 (shaded region OBCO) ( as it is symmetrical about the x - axis as well as 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 (0,2) 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 region enclosed by it is equal to square units.


11

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