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प्रश्न
Determine the area under the curve y = `sqrt("a"^2 - x^2)` included between the lines x = 0 and x = a.
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उत्तर
Here, we are given y = `sqrt("a"^2 - x^2)`
⇒ y2 = a2 – x2
⇒ x2 + y2 = a2
Area of the shaded region
= `2[(1)^(3/2) - 0] - 3/2[(1)^2 - 0]`
= `[x/2 sqrt("a"^2 - x^2) + "a"^2/2 sin^-1 x/"a"]_0^"a"`
= `["a"/2 sqrt("a"^2 - "a"^2) + "a"^2/2 sin^-1 "a"/"a" - 0 - 0]`
= `"a"^2/2 sin^-1 (1)`
= `"a"^2/2 * pi/2`
= `(pi"a"^2)/4`
Hence, the required area = `(pi"a"^2)/4` sq.units.
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