 ## 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: - We have to find the area of the shaded region.

Required area

= 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.

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