CBSE Class 12-science Answered

Integration

Asked by alanpeter9611 | 14 Feb, 2019, 05:01: PM

Expert Answer

Equation x² + y² = 1 represents circle with centre at origin and radius of 1 unit.

x² + y² ≤ 1 means point lying inside the circle.

x+y =1 is a line making equal intercept of 1 unit with slope = -1

x+y≥1 means points above the line.

hence region representing x² + y² ≤ 1 ≤ x+y is the shaded area shown in figure.

Area of the region =

begin mathsize 16px style integral subscript 0 superscript 1 square root of 1 minus x squared end root space d x space minus space integral subscript 0 superscript 1 open parentheses 1 minus x close parentheses d x space equals space I subscript 1 minus I subscript 2 end style

.......................(1)

begin mathsize 16px style w h e r e space I subscript 1 space equals space integral subscript 0 superscript 1 square root of 1 minus x squared end root space d x space space l e t space u s space s u b s t i t u t e space x space equals space sin theta comma space space d x space equals space cos theta d theta l i m i t s space colon negative space space w h e n space x space equals 0 comma space theta equals 0 comma space space space w h e n space x space equals space 1 comma space theta equals straight pi over 2 I subscript 1 space equals space integral subscript 0 superscript bevelled straight pi over 2 end superscript space cos squared theta d theta space equals space straight pi over 4 I subscript 2 space equals space integral subscript 0 superscript 1 left parenthesis 1 minus x right parenthesis d x space equals space 1 half end style

By substituting values of definite integrals I₁ and I₂ in eqn.(1), we get the area of shaded region as (π-2)/4 sq.units

Answered by Thiyagarajan K | 14 Feb, 2019, 06:33: PM

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