Question

A rectangle is bounded by the x-axis and the semicircle y=
`\sqrt{36-x^(2) }` (y =square root of 36-xsquare) Write the area of the rectangle as a function of x and determine the domain of the function

Answer 1

`x = + - √((6 + y)(6 - y))`

Explanation:

`1. \: y = \sqrt{6 {}^(2) - x {}^(2)} \\ 2. \: y = √((6 + x)(6 - x)) \\ 3. \: y {}^(2) = √((6 + x)(6 - x)) \\ 4. \: y {}^(2) = 6 {}^(2) - x {}^(2) \\ 5. \: y {}^(2) = 36 - x {}^(2) \\ 6. \: y {}^(2) - 36 = - x {}^(2) \\ 7. \: - y {}^(2) + 36 = x {}^(2) \\ 8. \: 36 - y {}^(2) = x {}^(2) \\ 9. \: + - \sqrt{36 - y {}^(2)} = x {}^(2) \\ 10. \: + - \sqrt{6 {}^(2) - y {}^(2) } = x {}^(2) \\ 11. \: + - √((6 + y)(6 - y)) = x \\ 12. \: x = + - √((6 + y)(6 - y))`