Answer:
The area is [tex]\frac{567}{8}u^2[/tex]
Step-by-step explanation:
The area of a flat region bounded by the graphs of two functions f (x) and g (x), with f (x)> g (x) can be found through the integral:
[tex]\int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
The integration limits are given by the intersection points of the graphs of the functions in the first quadrant. Then, the cut points are:
[tex]g(x) = x\\f(x) = 2x\sqrt{25-x^2}[/tex]
[tex]x=2x\sqrt{25-x^2}\\x^2=4x^2(25-x^2)\\x^2(1-100+4x^2)=0\\x_1=0\\x_2=\frac{3\sqrt{11}}{2}[/tex]
The area of the region is:
[tex]\int\limits^b_a {[f(x) - g(x)]} \, dx = \int\limits^{\frac{3\sqrt{11}}{2}}_0 {x(2\sqrt{25-x^2}-1)} \, dx = \frac{567}{8}u^2[/tex]
