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Solve the following : Find the area of the circle x2 + y2 = 9, using integration. - Mathematics and Statistics

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Sum

Solve the following :

Find the area of the circle x2 + y2 = 9, using integration.

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Solution

By the symmetry of the circle, its area is equal to 4 times the area of the region OABO. Clearly for this region, the limits of integration are 0 and 3.

From the equation of the circle, y2 = 9 – x2.
In the first quadrant, y > 0
∴ y = `sqrt(9 - x^2)`
∴ area of the circle = 4  (area of the region OABO)

= `4int_0^3y*dx = 4int_0^3 sqrt(9 - x^2)*dx`

= `4[x/2 sqrt(9 - x^2) + (9)/(2) sin^-1 (x/3)]_0^3`

= `4[3/2 sqrt(9 - 9) + (9)/(2) sin^-1 (3/3)] - 4[(0)/(2) sqrt(9 - 0) + (9)/(2)sin^1 (0)]`

= `4*(9)/(2)*(pi)/(2)`

= 9π sq units.

Concept: Application of Definite Integration
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 5 Application of Definite Integration
Miscellaneous Exercise 5 | Q 2.02 | Page 190
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