What is the area between the x-axis and the curve y = -(x- 5)^2 + 4

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The area enclosed by the curve `y = -(x- 5)^2 + 4` and the x-axis has to be determined.

The curve `y = -(x- 5)^2 + 4` intersects the x-axis at the points that are the solution of `-(x- 5)^2 + 4 = 0`

=> `(x - 5)^2 = 4`

=> `x - 5 = +-2`

=> x = 3 and x = 7

The required area is the integral `int_3^7 -(x- 5)^2 + 4 dx`

=> `int_3^7 -(x^2 + 25 - 10x) + 4 dx`

=> `int_3^7 -x^2 - 25 + 10x + 4 dx`

=> `int_3^7 -x^2 - 21 + 10x dx`

=> `[-x^3/3 - 21x + 10x^2/2]_3^7`

=> `(-1/3)*(7^3 - 3^3) + 5(7^2 - 3^2) - 21(7 - 3)`

=> `(-1/3)*316 + 5*40 - 21*4`

=> `32/3`

**The area between the x-axis and the curve `y = -(x- 5)^2 + 4` is `32/3` square units.**

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